Notation by work
Inverse Galois: source and evidence
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276 symbol spellings in 890 mathematical passages across 10 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 276 entries are available below.
dDefinition passage · 18 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
EDefinition passage · 72 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
\QDefinition passage · 192 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
tDefinition passage · 43 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
t_1Definition passage · 5 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
t_dDefinition passage · 5 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
\ldotsDefinition passage · 7 passages
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Definition passage · chapters01_03.tex, line 382
An extension E/Q(t_1,…,t_d) is regular over Q if every element of E algebraic over Q belongs to Q. In the characteristic zero setting considered here, this is the constant field condition used for regular finite Galois extensions.
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E/\Q(t_1,\ldots,t_d)
+Introduction passage · 33 passages
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Introduction passage · chapters01_03.tex, line 213
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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\zeta^2+\zeta+1=0.
=Introduction passage · 223 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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E=\Q(x_g:g\in G)
AIntroduction passage · 11 passages
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Introduction passage · chapters01_03.tex, line 385
Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A monic minimal polynomial m∈Q[T] satisfies m(A/B)=0 identically. Evaluating at that tuple gives a rational root of m. Irreducibility forces m …
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r=A/B
aIntroduction passage · 38 passages
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Introduction passage · chapters04_06.tex, line 11
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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\lambda(a)\delta_g=\delta_{ag}. \label{eq:b-regular-action}
BIntroduction passage · 4 passages
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Introduction passage · chapters01_03.tex, line 385
Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A monic minimal polynomial m∈Q[T] satisfies m(A/B)=0 identically. Evaluating at that tuple gives a rational root of m. Irreducibility forces m …
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r=A/B
\CIntroduction passage · 8 passages
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Introduction passage · chapters04_06.tex, line 10
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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V=\C[G]
cIntroduction passage · 16 passages
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Introduction passage · chapters04_06.tex, line 16
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
Exact mathematical passage
\lambda\!\left(\sum_{g\in G}c_g g\right)\delta_1 =\sum_{g\in G}c_g\delta_g. \label{eq:b-regular-faithfulness}
c_gIntroduction passage · 2 passages
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Introduction passage · chapters04_06.tex, line 16
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
Exact mathematical passage
\lambda\!\left(\sum_{g\in G}c_g g\right)\delta_1 =\sum_{g\in G}c_g\delta_g. \label{eq:b-regular-faithfulness}
eIntroduction passage · 17 passages
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Introduction passage · chapters04_06.tex, line 79
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
Exact mathematical passage
|HK|=\frac{|H||K|}{|H\cap K|}, \qquad e_He_K=\frac1{|HK|}\sum_{x\in HK}x. \label{eq:b-product-support}
E^GIntroduction passage · 2 passages
Search aliases: E^G
Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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K=E^G
e_HIntroduction passage · 6 passages
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Introduction passage · chapters04_06.tex, line 79
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
Exact mathematical passage
|HK|=\frac{|H||K|}{|H\cap K|}, \qquad e_He_K=\frac1{|HK|}\sum_{x\in HK}x. \label{eq:b-product-support}
e_KIntroduction passage · 3 passages
Search aliases: e_K
Introduction passage · chapters04_06.tex, line 79
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
Exact mathematical passage
|HK|=\frac{|H||K|}{|H\cap K|}, \qquad e_He_K=\frac1{|HK|}\sum_{x\in HK}x. \label{eq:b-product-support}
FIntroduction passage · 17 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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F
fIntroduction passage · 33 passages
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Introduction passage · chapters01_03.tex, line 212
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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f(X)=X^3-2
GIntroduction passage · 119 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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G
gIntroduction passage · 51 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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E=\Q(x_g:g\in G)
HIntroduction passage · 119 passages
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Introduction passage · chapters04_06.tex, line 78
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
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HK
hIntroduction passage · 38 passages
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Introduction passage · chapters04_06.tex, line 78
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
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hk
KIntroduction passage · 86 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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K=E^G
kIntroduction passage · 21 passages
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Introduction passage · chapters04_06.tex, line 78
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
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hk
LIntroduction passage · 78 passages
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Introduction passage · chapters01_03.tex, line 217
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
Exact mathematical passage
L=\Q(\alpha,\zeta).
mIntroduction passage · 5 passages
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Introduction passage · chapters01_03.tex, line 385
nal function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A monic minimal polynomial m∈Q[T] satisfies m(A/B)=0 identically. Evaluating at that tuple gives a rational root of m. Irreducibility forces m to have d…
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m\in\Q[T]
rIntroduction passage · 35 passages
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Introduction passage · chapters01_03.tex, line 385
Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A monic minimal polynomial m∈Q[T] satisfies m(A/B)=0 identically. Evaluating at that tuple gives a rational root of m. Irreducibility forces m …
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r\in E
TIntroduction passage · 9 passages
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Introduction passage · chapters01_03.tex, line 385
nal function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A monic minimal polynomial m∈Q[T] satisfies m(A/B)=0 identically. Evaluating at that tuple gives a rational root of m. Irreducibility forces m to have d…
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m\in\Q[T]
VIntroduction passage · 19 passages
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Introduction passage · chapters04_06.tex, line 10
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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V=\C[G]
XIntroduction passage · 41 passages
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Introduction passage · chapters01_03.tex, line 212
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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f(X)=X^3-2
xIntroduction passage · 41 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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E=\Q(x_g:g\in G)
X^3Introduction passage · 13 passages
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Introduction passage · chapters01_03.tex, line 212
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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f(X)=X^3-2
x_gIntroduction passage · 10 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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E=\Q(x_g:g\in G)
|Introduction passage · 33 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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[E:K]=|G|
\alphaIntroduction passage · 47 passages
Search aliases: α alpha
Introduction passage · chapters01_03.tex, line 212
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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\alpha
\deltaIntroduction passage · 13 passages
Search aliases: δ delta
Introduction passage · chapters04_06.tex, line 10
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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\{\delta_g:g\in G\}
\delta_1Introduction passage · 4 passages
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Introduction passage · chapters04_06.tex, line 16
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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\lambda\!\left(\sum_{g\in G}c_g g\right)\delta_1 =\sum_{g\in G}c_g\delta_g. \label{eq:b-regular-faithfulness}
\delta_{ag}Introduction passage · 1 passage
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Introduction passage · chapters04_06.tex, line 11
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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\lambda(a)\delta_g=\delta_{ag}. \label{eq:b-regular-action}
\delta_gIntroduction passage · 3 passages
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Introduction passage · chapters04_06.tex, line 10
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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\{\delta_g:g\in G\}
\zetaIntroduction passage · 23 passages
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Introduction passage · chapters01_03.tex, line 212
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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\zeta
\zeta^2Introduction passage · 3 passages
Search aliases: ζ^2 zeta
Introduction passage · chapters01_03.tex, line 213
Take f(X)=X^3-2 and let α denote its positive real root. Eisenstein's criterion at the prime two proves that f is irreducible. Equivalently, a rational root would have to be an integer divisor of two, and none has cube two. Let ζ be a primitive cube root of unity, so ζ^2+ζ+1=0. The three roots are α,ζα,ζ^2α. Their splitting field is L=Q(α,ζ). Indeed, this field contains every root, and the ratio of the second root to the first recovers ζ from the splitting field.
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\zeta^2+\zeta+1=0.
\lambdaIntroduction passage · 18 passages
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Introduction passage · chapters04_06.tex, line 11
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
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\lambda(a)\delta_g=\delta_{ag}. \label{eq:b-regular-action}
\inIntroduction passage · 52 passages
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Introduction passage · appendices.tex, line 6
The symbol G denotes a prescribed finite group, except where explicitly identified with the full Galois group of an existing extension. The arithmetic base is Q. The field E=Q(x_g:g∈ G) in Chapter~[source reference] is also written F in the visual plates. Its invariant field is K=E^G. The equality [E:K]=|G| describes a finite extension over K, while E/Q has positive transcendence degree.
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E=\Q(x_g:g\in G)
\sumIntroduction passage · 11 passages
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Introduction passage · chapters04_06.tex, line 16
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
Exact mathematical passage
\lambda\!\left(\sum_{g\in G}c_g g\right)\delta_1 =\sum_{g\in G}c_g\delta_g. \label{eq:b-regular-faithfulness}
\sum_{g\in G}Introduction passage · 2 passages
Search aliases: ∑_g∈ G sum in
Introduction passage · chapters04_06.tex, line 16
Let G be a finite group. Write V=C[G] for the vector space with orthonormal basis {δ_g:g∈ G}. The left regular action is λ(a)δ_g=δ_ag. Every λ(a) is unitary. Extend λ linearly to the group algebra. This extension is injective, since λ (∑_g∈ Gc_g g)δ_1 =∑_g∈ Gc_gδ_g. Equality of regular operators therefore gives equality of all group algebra coefficients. This stronger property will justify the reverse implications below.
Exact mathematical passage
\lambda\!\left(\sum_{g\in G}c_g g\right)\delta_1 =\sum_{g\in G}c_g\delta_g. \label{eq:b-regular-faithfulness}
\sum_{x\in HK}Introduction passage · 1 passage
Search aliases: ∑_x∈ HK sum in
Introduction passage · chapters04_06.tex, line 79
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
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|HK|=\frac{|H||K|}{|H\cap K|}, \qquad e_He_K=\frac1{|HK|}\sum_{x\in HK}x. \label{eq:b-product-support}
\capIntroduction passage · 9 passages
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Introduction passage · chapters04_06.tex, line 78
[Uniform product multiplicity] Every element of HK has exactly |H∩ K| expressions as hk. Hence |HK|=frac|H||K||H∩ K|, e_He_K=frac1|HK|∑_x∈ HKx. The last sum is defined even when HK is not a subgroup.
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|H\cap K|
\neIntroduction passage · 4 passages
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Introduction passage · chapters01_03.tex, line 385
Our rational function field E=Q(x_g) has this property. To see the constant field assertion directly, suppose r∈ E is algebraic over Q. Write r=A/B for polynomials over Q with B≠0. There is a rational tuple at which B is nonzero: a nonzero polynomial over the infinite field Q cannot vanish on every rational tuple, by induction on the number of variables. A monic minimal polynomial m∈Q[T] satisfies m(A/B)=0 identically. Evaluating at that tuple gives a rational root of m. Irreducibility forces m …
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B\ne0
\geIntroduction passage · 2 passages
Search aliases: ≥ ge
Introduction passage · chapters01_03.tex, line 389
[Hilbert specialization, external input] Let d≥1 and let E/Q(t_1,…,t_d) be a finite regular Galois extension with group G. There is a Hilbert subset of rational parameter tuples for which specialization is defined and the specialized extension has full Galois group G. The tuples may be required to avoid the zero set of any prescribed nonzero polynomial that records finitely many excluded denominators or discriminants.
Exact mathematical passage
d\ge1
*Use in source · 2 passages
Search aliases: *
Use in source · chapters04_06.tex, line 47
Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed vector is unchanged by every summand, and therefore by the average.
Exact mathematical passage
E_H^*=E_H
a^{-1}Use in source · 1 passage
Search aliases: a^-1
Use in source · chapters04_06.tex, line 89
Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.
Exact mathematical passage
(h_0a,a^{-1}k_0)
a^2Use in source · 5 passages
Search aliases: a^2
Use in source · chapters01_03.tex, line 94
Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.
Exact mathematical passage
a^2=2b^2
a^3Use in source · 1 passage
Search aliases: a^3
Use in source · chapters01_03.tex, line 149
For a monic cubic X^3+aX^2+bX+c, the discriminant is a^2b^2-4b^3-4a^3c-27c^2+18abc. Here this gives disc(f)=4+32+4-27+36=49. The discriminant is nonzero, so the roots are distinct.
Exact mathematical passage
a^2b^2-4b^3-4a^3c-27c^2+18abc.
A_3Use in source · 2 passages
Search aliases: A_3
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
A_3
A_4Use in source · 2 passages
Search aliases: A_4
Use in source · chapters01_03.tex, line 276
roots. Its order is divisible by four and three, so it is divisible by twelve. Since H≤ S_4, its order is twelve or twenty four. A subgroup of order twelve would have index two and would be normal. The quotient map S_4→ C_2 would send all transpositions to the same element because they are conjugate. They generate S_4, so this element must be nontrivial. Its kernel is therefore exactly A_4, the permutations expressible as a product of an even number of transpositions. But a four cycle is odd. Si…
Exact mathematical passage
A_4
a_gUse in source · 1 passage
Search aliases: a_g
Use in source · chapters07_08.tex, line 68
The normal basis theorem supplies a regular module for an already existing finite Galois extension [source reference]. If L/Q has group G and normal basis generator α, the vector-space map Φ:Q[G]⟶ L, ∑_g a_g g⟼∑_g a_g g(α) is an equivariant linear isomorphism. It transfers subgroup averages and their fixed vector spaces. It is not a claim that the usual multiplication in Q[G] is field multiplication.
Exact mathematical passage
\Phi:\Q[G]\longrightarrow L, \qquad \sum_g a_g g\longmapsto\sum_g a_g g(\alpha)
\AutUse in source · 2 passages
Search aliases: Aut
Use in source · chapters01_03.tex, line 319
[Artin's finite automorphism theorem, external input] Let E be a field and A a finite subgroup of Aut(E). If K=E^A, then E/K is a finite Galois extension, [E:K]=|A|, Gal(E/K)=A.
Exact mathematical passage
\Aut(E)
\Aut_{\Q}Use in source · 1 passage
Search aliases: Aut_Q Aut Q
Use in source · chapters07_08.tex, line 38
The substitutions give a homomorphism from G into Aut_Q(L). The identity and composition congruences prove the homomorphism law. Distinct generator images make the homomorphism injective. Write its image as Σ, so |Σ|=n.
Exact mathematical passage
\Aut_{\Q}(L)
bUse in source · 25 passages
Search aliases: b
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
b^2Use in source · 6 passages
Search aliases: b^2
Use in source · chapters01_03.tex, line 94
Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.
Exact mathematical passage
a^2=2b^2
b^3Use in source · 1 passage
Search aliases: b^3
Use in source · chapters01_03.tex, line 149
For a monic cubic X^3+aX^2+bX+c, the discriminant is a^2b^2-4b^3-4a^3c-27c^2+18abc. Here this gives disc(f)=4+32+4-27+36=49. The discriminant is nonzero, so the roots are distinct.
Exact mathematical passage
a^2b^2-4b^3-4a^3c-27c^2+18abc.
\bmodUse in source · 3 passages
Search aliases: bmod
Use in source · chapters04_06.tex, line 432
[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.
Exact mathematical passage
\Gamma=\Z/2\Z, \quad E=\Z/4\Z, \quad G=\Z/2\Z, \quad \pi(x)=x\bmod2, \quad \rho=\operatorname{id}_{\Gamma}.
\bottomruleUse in source · 1 passage
Search aliases: bottomrule
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
CUse in source · 24 passages
Search aliases: C
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
c'Use in source · 1 passage
Search aliases: c'
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
c^2Use in source · 2 passages
Search aliases: c^2
Use in source · chapters01_03.tex, line 94
Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.
Exact mathematical passage
b^2=2c^2
C_2Use in source · 18 passages
Search aliases: C_2
Use in source · chapters01_03.tex, line 92
Quadratic symmetry: C_2.
Exact mathematical passage
C_2
C_3Use in source · 3 passages
Search aliases: C_3
Use in source · chapters01_03.tex, line 58
Transitivity does useful work. The orbit of a root has size d, so the order of the group is divisible by d. It does not identify the group by itself. A cubic can have group C_3 or S_3. A quartic can have several transitive subgroups of S_4. The examples in the next chapter show how extra information selects the group.
Exact mathematical passage
C_3
C_4Use in source · 3 passages
Search aliases: C_4
Use in source · chapters04_06.tex, line 429
[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.
Exact mathematical passage
C_4
d_1Use in source · 2 passages
Search aliases: d_1
Use in source · chapters01_03.tex, line 249
[Good prime cycle theorem, external input] Let f∈Z[X] be monic, and suppose its reduction modulo a prime p is squarefree. If its irreducible factors over F_p have degrees d_1,…,d_r, the Galois group of its splitting field over Q, acting on the roots of f, contains a permutation with disjoint cycle lengths d_1,…,d_r.
Exact mathematical passage
d_1,\ldots,d_r
d_rUse in source · 2 passages
Search aliases: d_r
Use in source · chapters01_03.tex, line 249
[Good prime cycle theorem, external input] Let f∈Z[X] be monic, and suppose its reduction modulo a prime p is squarefree. If its irreducible factors over F_p have degrees d_1,…,d_r, the Galois group of its splitting field over Q, acting on the roots of f, contains a permutation with disjoint cycle lengths d_1,…,d_r.
Exact mathematical passage
d_1,\ldots,d_r
\degUse in source · 1 passage
Search aliases: deg
Use in source · chapters07_08.tex, line 45
Apply the finite automorphism fixed-field theorem, in the form recorded in source contract LC10. It gives a finite Galois extension L/L^Σ with [L:L^Σ]=|Σ|=n, Gal(L/L^Σ)=Σ. All substitutions fix Q, hence Q⊆ L^Σ. Since [L:Q]=deg f=n, the degree formula gives n=[L:L^Σ][L^Σ:Q] =n[L^Σ:Q]. Therefore L^Σ=Q. The preceding Galois assertion now has the required rational base, and the full group is Σ≃ G.
Exact mathematical passage
[L:\Q]=\deg f=n
\operatorname{disc}Use in source · 1 passage
Search aliases: disc operatorname
Use in source · chapters01_03.tex, line 153
For a monic cubic X^3+aX^2+bX+c, the discriminant is a^2b^2-4b^3-4a^3c-27c^2+18abc. Here this gives disc(f)=4+32+4-27+36=49. The discriminant is nonzero, so the roots are distinct.
Exact mathematical passage
\label{eq:g02-cubic-disc} \operatorname{disc}(f)=4+32+4-27+36=49.
E^AUse in source · 1 passage
Search aliases: E^A
Use in source · chapters01_03.tex, line 319
[Artin's finite automorphism theorem, external input] Let E be a field and A a finite subgroup of Aut(E). If K=E^A, then E/K is a finite Galois extension, [E:K]=|A|, Gal(E/K)=A.
Exact mathematical passage
K=E^A
e_{gh}Use in source · 1 passage
Search aliases: e_gh
Use in source · chapters01_03.tex, line 74
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
\rho(g)e_h=e_{gh}.
e_{gHg^{-1}}Use in source · 1 passage
Search aliases: e_gHg^-1
Use in source · chapters04_06.tex, line 187
Conjugation gives ge_Hg^-1=e_gHg^-1. Equality of these group algebra elements is equality of their supports, hence equality of the subgroups. This proves the equivalence of the first two statements. Injectivity of λ proves equivalence with the third.
Exact mathematical passage
ge_Hg^{-1}=e_{gHg^{-1}}
e_{gx}Use in source · 2 passages
Search aliases: e_gx
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
\lambda(g)e_x=e_{gx}
E_HUse in source · 18 passages
Search aliases: E_H
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
E_H=|H|^{-1}\sum_{h\in H}\lambda(h)
e_hUse in source · 3 passages
Search aliases: e_h
Use in source · chapters01_03.tex, line 73
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
e_h
E_H^*Use in source · 1 passage
Search aliases: E_H^*
Use in source · chapters04_06.tex, line 47
Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed vector is unchanged by every summand, and therefore by the average.
Exact mathematical passage
E_H^*=E_H
E_H^2Use in source · 1 passage
Search aliases: E_H^2
Use in source · chapters04_06.tex, line 48
Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed vector is unchanged by every summand, and therefore by the average.
Exact mathematical passage
E_H^2=\frac1{|H|^2}\sum_{h,k\in H}\lambda(hk) =\frac1{|H|}\sum_{u\in H}\lambda(u)=E_H.
E_{HK}Use in source · 1 passage
Search aliases: E_HK
Use in source · chapters04_06.tex, line 99
[Exact compatibility criterion] In the regular realization, the following statements are equivalent: HK=KH, HK is a subgroup, E_HE_K=E_KE_H. When they hold, E_HE_K=E_KE_H=E_HK=E_⟨ H,K⟩. Without a compatibility assumption, the intersection of the fixed ranges is still V^H∩ V^K=V^⟨ H,K⟩.
Exact mathematical passage
E_HE_K=E_KE_H=E_{HK}=E_{\langle H,K\rangle}. \label{eq:b-compatible-product}
e_{HK}Use in source · 1 passage
Search aliases: e_HK
Use in source · chapters04_06.tex, line 113
For permutable subgroups, [source reference] gives e_He_K=e_HK=e_Ke_H. Applying λ proves the operator equality. If the operators commute, injectivity in [source reference] gives e_He_K=e_Ke_H. Their supports are HK and KH, with positive coefficients, so the sets are equal. A subgroup HK contains both H and K, and is contained in every subgroup containing both, hence equals ⟨ H,K⟩. Finally, a vector fixed by both groups is fixed by every word in their elements and inverses. This proves the inters…
Exact mathematical passage
e_He_K=e_{HK}=e_Ke_H
E_KUse in source · 8 passages
Search aliases: E_K
Use in source · appendices.tex, line 10
Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.
Exact mathematical passage
\operatorname{ran}(E_K)\subseteq\operatorname{ran}(E_H)
e_kUse in source · 1 passage
Search aliases: e_k
Use in source · chapters01_03.tex, line 79
They do not yet give an extension of Q. The vector space Q^G has no designated field multiplication. Its coordinatewise multiplication makes it a product algebra, which has zero divisors whenever |G|>1. For distinct h,k, the coordinate idempotents satisfy e_he_k=0 although both are nonzero. A finite product of copies of Q is a useful algebraic symmetry model, but it cannot be the required nontrivial number field.
Exact mathematical passage
e_he_k=0
e_xUse in source · 2 passages
Search aliases: e_x
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
\lambda(g)e_x=e_{gx}
E_{\{1\}}Use in source · 1 passage
Search aliases: E_{1}
Use in source · chapters04_06.tex, line 61
In particular, the identity subgroup gives E_{1}=I, while the whole group gives a rank one projector onto the constant vector. Every intermediate subgroup supplies a resolution {E_H,I-E_H}. The word resolution here refers to the vector space decomposition into fixed vectors and their orthogonal complement. A field structure has not been imposed on these ranges.
Exact mathematical passage
E_{\{1\}}=I
E_{\langle H,K\rangle}Use in source · 1 passage
Search aliases: E_⟨ H,K⟩ langle rangle
Use in source · chapters04_06.tex, line 99
[Exact compatibility criterion] In the regular realization, the following statements are equivalent: HK=KH, HK is a subgroup, E_HE_K=E_KE_H. When they hold, E_HE_K=E_KE_H=E_HK=E_⟨ H,K⟩. Without a compatibility assumption, the intersection of the fixed ranges is still V^H∩ V^K=V^⟨ H,K⟩.
Exact mathematical passage
E_HE_K=E_KE_H=E_{HK}=E_{\langle H,K\rangle}. \label{eq:b-compatible-product}
\mathbf FUse in source · 4 passages
Search aliases: F mathbf
Use in source · chapters01_03.tex, line 243
For a quartic realization, consider f(X)=X^4-X-1. Its reduction modulo two is X^4+X+1. It has no root in F_2. The only monic irreducible quadratic over F_2 is X^2+X+1. Modulo that polynomial, X^2=X+1, X^3=1 and X^4=X, so X^4+X+1 has remainder one. A reducible quartic without a linear factor must factor as two irreducible quadratics. This possibility has just been excluded, proving that the reduction is irreducible. Gauss's lemma now proves that f is irreducible over Q.
Exact mathematical passage
\mathbf F_2
F^GUse in source · 1 passage
Search aliases: F^G
Use in source · chapters01_03.tex, line 375
[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/09_invariant_base.pdf captionThe invariant field contains rational constants and also transcendental invariants. The diagram writes F=E and K=F^G. Its transcendence degree is fixed by the finite extension E/K. Pure transcendence requires an additional rationality argument before Hilbert specialization can be applied through a rational parameter presentation.
Exact mathematical passage
K=F^G
\mathbf F_2Use in source · 2 passages
Search aliases: F_2 mathbf
Use in source · chapters01_03.tex, line 243
For a quartic realization, consider f(X)=X^4-X-1. Its reduction modulo two is X^4+X+1. It has no root in F_2. The only monic irreducible quadratic over F_2 is X^2+X+1. Modulo that polynomial, X^2=X+1, X^3=1 and X^4=X, so X^4+X+1 has remainder one. A reducible quartic without a linear factor must factor as two irreducible quadratics. This possibility has just been excluded, proving that the reduction is irreducible. Gauss's lemma now proves that f is irreducible over Q.
Exact mathematical passage
\mathbf F_2
\mathbf F_7Use in source · 1 passage
Search aliases: F_7 mathbf
Use in source · chapters01_03.tex, line 255
Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6
Exact mathematical passage
\label{eq:g02-mod-seven} f(X)=(X-3)(X^3+3X^2+2X+5) \quad\text{in }\mathbf F_7[X].
\mathbf F_pUse in source · 1 passage
Search aliases: F_p mathbf
Use in source · chapters01_03.tex, line 249
[Good prime cycle theorem, external input] Let f∈Z[X] be monic, and suppose its reduction modulo a prime p is squarefree. If its irreducible factors over F_p have degrees d_1,…,d_r, the Galois group of its splitting field over Q, acting on the roots of f, contains a permutation with disjoint cycle lengths d_1,…,d_r.
Exact mathematical passage
\mathbf F_p
f_tUse in source · 1 passage
Search aliases: f_t
Use in source · chapters04_06.tex, line 339
The polynomial f_t(X)=X^3-t is a small exact example of what can happen under specialization. It is used here to study changes in the splitting field group. Its generic splitting field is not a regular extension of Q(t), a fact established below.
Exact mathematical passage
f_t(X)=X^3-t \label{eq:b-cubic-family}
g^{-1}Use in source · 1 passage
Search aliases: g^-1
Use in source · chapters04_06.tex, line 187
Conjugation gives ge_Hg^-1=e_gHg^-1. Equality of these group algebra elements is equality of their supports, hence equality of the subgroups. This proves the equivalence of the first two statements. Injectivity of λ proves equivalence with the third.
Exact mathematical passage
ge_Hg^{-1}=e_{gHg^{-1}}
g_1Use in source · 1 passage
Search aliases: g_1
Use in source · chapters01_03.tex, line 77
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
\rho(g_1g_2)=\rho(g_1)\rho(g_2)
g_2Use in source · 1 passage
Search aliases: g_2
Use in source · chapters01_03.tex, line 77
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
\rho(g_1g_2)=\rho(g_1)\rho(g_2)
G_{\Q}Use in source · 9 passages
Search aliases: G_Q Q
Use in source · appendices.tex, line 12
An arithmetic embedding problem uses the profinite group G_Q=Gal(overlineQ/Q) with its Krull topology. A finite target group has the discrete topology. The lift is continuous, compatible with the prescribed quotient, and proper when surjective. The finite cyclic examples in Chapter~6 use a different ambient group and are explicitly models.
Exact mathematical passage
G_{\Q}=\Gal(\overline{\Q}/\Q)
\GalUse in source · 15 passages
Search aliases: Gal
Use in source · appendices.tex, line 12
An arithmetic embedding problem uses the profinite group G_Q=Gal(overlineQ/Q) with its Krull topology. A finite target group has the discrete topology. The lift is continuous, compatible with the prescribed quotient, and proper when surjective. The finite cyclic examples in Chapter~6 use a different ambient group and are explicitly models.
Exact mathematical passage
G_{\Q}=\Gal(\overline{\Q}/\Q)
\operatorname{GL}Use in source · 1 passage
Search aliases: GL operatorname
Use in source · chapters01_03.tex, line 77
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
G\hookrightarrow\operatorname{GL}(\Q^G)
h^{-1}Use in source · 1 passage
Search aliases: h^-1
Use in source · chapters01_03.tex, line 300
For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\sigma_{h^{-1}}
h_0Use in source · 3 passages
Search aliases: h_0
Use in source · chapters04_06.tex, line 89
Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.
Exact mathematical passage
x=h_0k_0
h_0^{-1}Use in source · 1 passage
Search aliases: h_0^-1
Use in source · chapters04_06.tex, line 89
Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.
Exact mathematical passage
h_0^{-1}h=k_0k^{-1}\in H\cap K
\hlineUse in source · 1 passage
Search aliases: hline
Use in source · chapters01_03.tex, line 261
Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreducible. Its derivative 3X^2+6X+2 is nonzero and has smaller degree, so it cannot have a common factor with this irreducible cubic. The cubic's value at t…
Exact mathematical passage
\begin{array}{c|rrrrrrr} x&0&1&2&3&4&5&6\\\hline x^3+3x^2+2x+5\pmod7&5&4&1&2&6&5&5. \end{array}
IUse in source · 3 passages
Search aliases: I
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
iUse in source · 5 passages
Search aliases: i
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\operatorname{id}Use in source · 1 passage
Search aliases: id operatorname
Use in source · chapters04_06.tex, line 432
[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.
Exact mathematical passage
\Gamma=\Z/2\Z, \quad E=\Z/4\Z, \quad G=\Z/2\Z, \quad \pi(x)=x\bmod2, \quad \rho=\operatorname{id}_{\Gamma}.
\operatorname{id}_{\Gamma}Use in source · 1 passage
Search aliases: id_Γ operatorname Gamma
Use in source · chapters04_06.tex, line 432
[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.
Exact mathematical passage
\Gamma=\Z/2\Z, \quad E=\Z/4\Z, \quad G=\Z/2\Z, \quad \pi(x)=x\bmod2, \quad \rho=\operatorname{id}_{\Gamma}.
JUse in source · 3 passages
Search aliases: J
Use in source · chapters04_06.tex, line 410
Let J=widetildeρ(G_Q)≤ E. Even a weak solution satisfies π(J)=G, since ρ is onto. The inclusion kerwidetildeρ⊆kerρ gives a finite Galois field M containing L. Its group is J, rather than necessarily E. Properness makes the exact arithmetic tower Gal(M/Q)≅ E, Gal(M/L)≅ N, Gal(L/Q)≅ G. The quotient map in this tower must be the specified π under these identifications.
Exact mathematical passage
J=\widetilde\rho(G_{\Q})\leq E
jUse in source · 1 passage
Search aliases: j
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
V=\prod_{i<j}(r_i-r_j)
k^{-1}Use in source · 1 passage
Search aliases: k^-1
Use in source · chapters04_06.tex, line 89
Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.
Exact mathematical passage
h_0^{-1}h=k_0k^{-1}\in H\cap K
k_0Use in source · 4 passages
Search aliases: k_0
Use in source · chapters04_06.tex, line 89
Fix one expression x=h_0k_0. An equality hk=h_0k_0 is equivalent to h_0^-1h=k_0k^-1∈ H∩ K. For each a∈ H∩ K, the pair (h_0a,a^-1k_0) gives one expression. This is a bijection. Counting all pairs proves the cardinality formula, and counting their coefficients in the group algebra proves the second identity.
Exact mathematical passage
x=h_0k_0
\kerUse in source · 5 passages
Search aliases: ker
Use in source · chapters04_06.tex, line 410
Let J=widetildeρ(G_Q)≤ E. Even a weak solution satisfies π(J)=G, since ρ is onto. The inclusion kerwidetildeρ⊆kerρ gives a finite Galois field M containing L. Its group is J, rather than necessarily E. Properness makes the exact arithmetic tower Gal(M/Q)≅ E, Gal(M/L)≅ N, Gal(L/Q)≅ G. The quotient map in this tower must be the specified π under these identifications.
Exact mathematical passage
\ker\widetilde\rho\subseteq\ker\rho
lUse in source · 1 passage
Search aliases: l
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
L^HUse in source · 13 passages
Search aliases: L^H
Use in source · chapters01_03.tex, line 63
[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/02_field_group.pdf captionFor an already supplied finite Galois extension L/K with group G, subgroups H≤ G correspond to intermediate fixed fields L^H. The direction of inclusion reverses. The classical correspondence computes degrees and organizes fields after the ambient field extension has been established [source reference].
Exact mathematical passage
L^H
L^{H\cap K}Use in source · 1 passage
Search aliases: L^H∩ K cap
Use in source · chapters04_06.tex, line 223
The lattice identities follow directly. An element fixed by both H and K is fixed by their generated subgroup, so L^H∩ L^K=L^⟨ H,K⟩. The automorphisms fixing the compositum L^HL^K are precisely those fixing each field, namely H∩ K. The correspondence then gives L^HL^K=L^H∩ K. These identities require no permutability assumption. The stronger product identity for two averaging operators does require it. A field lattice operation and an ordered composition of linear maps are different operations.
Exact mathematical passage
L^HL^K=L^{H\cap K}. \label{eq:b-field-compositum}
L^KUse in source · 6 passages
Search aliases: L^K
Use in source · chapters04_06.tex, line 218
The lattice identities follow directly. An element fixed by both H and K is fixed by their generated subgroup, so L^H∩ L^K=L^⟨ H,K⟩. The automorphisms fixing the compositum L^HL^K are precisely those fixing each field, namely H∩ K. The correspondence then gives L^HL^K=L^H∩ K. These identities require no permutability assumption. The stronger product identity for two averaging operators does require it. A field lattice operation and an ordered composition of linear maps are different operations.
Exact mathematical passage
L^H\cap L^K=L^{\langle H,K\rangle}. \label{eq:b-field-intersection}
L^\SigmaUse in source · 5 passages
Search aliases: L^Σ Sigma
Use in source · chapters07_08.tex, line 40
Apply the finite automorphism fixed-field theorem, in the form recorded in source contract LC10. It gives a finite Galois extension L/L^Σ with [L:L^Σ]=|Σ|=n, Gal(L/L^Σ)=Σ. All substitutions fix Q, hence Q⊆ L^Σ. Since [L:Q]=deg f=n, the degree formula gives n=[L:L^Σ][L^Σ:Q] =n[L^Σ:Q]. Therefore L^Σ=Q. The preceding Galois assertion now has the required rational base, and the full group is Σ≃ G.
Exact mathematical passage
L/L^\Sigma
L^{\langle H,K\rangle}Use in source · 1 passage
Search aliases: L^⟨ H,K⟩ langle rangle
Use in source · chapters04_06.tex, line 218
The lattice identities follow directly. An element fixed by both H and K is fixed by their generated subgroup, so L^H∩ L^K=L^⟨ H,K⟩. The automorphisms fixing the compositum L^HL^K are precisely those fixing each field, namely H∩ K. The correspondence then gives L^HL^K=L^H∩ K. These identities require no permutability assumption. The stronger product identity for two averaging operators does require it. A field lattice operation and an ordered composition of linear maps are different operations.
Exact mathematical passage
L^H\cap L^K=L^{\langle H,K\rangle}. \label{eq:b-field-intersection}
L^{\langle st\rangle}Use in source · 1 passage
Search aliases: L^⟨ st⟩ langle rangle
Use in source · chapters01_03.tex, line 136
The fixed fields can also be read directly from the basis. The elements fixed by s are a+csqrt3, so L^⟨ s⟩=Q(sqrt3). Similarly, L^⟨ t⟩=Q(sqrt2). The product st changes the signs of both square roots but fixes their product, giving L^⟨ st⟩=Q(sqrt6). An element fixed by both s and t has only its rational component left.
Exact mathematical passage
L^{\langle st\rangle}=\Q(\sqrt6)
L^{\langle s\rangle}Use in source · 1 passage
Search aliases: L^⟨ s⟩ langle rangle
Use in source · chapters01_03.tex, line 136
The fixed fields can also be read directly from the basis. The elements fixed by s are a+csqrt3, so L^⟨ s⟩=Q(sqrt3). Similarly, L^⟨ t⟩=Q(sqrt2). The product st changes the signs of both square roots but fixes their product, giving L^⟨ st⟩=Q(sqrt6). An element fixed by both s and t has only its rational component left.
Exact mathematical passage
L^{\langle s\rangle}=\Q(\sqrt3)
L^{\langle t\rangle}Use in source · 1 passage
Search aliases: L^⟨ t⟩ langle rangle
Use in source · chapters01_03.tex, line 136
The fixed fields can also be read directly from the basis. The elements fixed by s are a+csqrt3, so L^⟨ s⟩=Q(sqrt3). Similarly, L^⟨ t⟩=Q(sqrt2). The product st changes the signs of both square roots but fixes their product, giving L^⟨ st⟩=Q(sqrt6). An element fixed by both s and t has only its rational component left.
Exact mathematical passage
L^{\langle t\rangle}=\Q(\sqrt2)
L^{\langle\sigma\rangle}Use in source · 1 passage
Search aliases: L^⟨σ⟩ langle sigma rangle
Use in source · chapters01_03.tex, line 106
The polynomial already splits in L, so L/Q is Galois and Gal(L/Q)={1,σ}≃ C_2. Its fixed elements satisfy a+bα=a-bα, whence b=0. This directly checks L^⟨σ⟩=Q without appealing to an abstract correspondence. The same argument works for every nonsquare d∈Q^×, with sqrt2 replaced by sqrt d.
Exact mathematical passage
L^{\langle\sigma\rangle}=\Q
L_2Use in source · 1 passage
Search aliases: L_2
Use in source · chapters04_06.tex, line 346
At t=2, the polynomial X^3-2 is irreducible by Eisenstein at the prime 2. Let α=sqrt[3]2 be the real root and let ζ be a primitive cube root of unity. The splitting field is L_2=Q(α,ζ). The real field Q(α) has degree three. It cannot contain the nonreal element ζ. Adjoining ζ, whose minimal polynomial is X^2+X+1, therefore doubles the degree. The splitting field has degree six and its faithful action on the three roots identifies its group with S_3. Equivalently, the discriminant is -27·2^2=-108…
Exact mathematical passage
L_2=\Q(\alpha,\zeta).
MUse in source · 6 passages
Search aliases: M
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
M_{23}Use in source · 4 passages
Search aliases: M_23
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\midUse in source · 1 passage
Search aliases: mid
Use in source · chapters01_03.tex, line 290
Fix a finite group G. For every g∈ G, introduce an independent variable x_g. Let E=Q(x_gmid g∈ G) be the rational function field in these |G| variables. Algebraic independence means that no nonzero polynomial with rational coefficients in the x_g vanishes. In particular distinct variables are distinct elements of E.
Exact mathematical passage
E=\Q(x_g\mid g\in G)
NUse in source · 20 passages
Search aliases: N
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\operatorname{N}Use in source · 3 passages
Search aliases: N operatorname
Use in source · chapters04_06.tex, line 248
In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs a nonlinear transfer.
Exact mathematical passage
\operatorname{N}_{L/L^H}(x)=\prod_{h\in H}h(x). \label{eq:b-norm-product}
nUse in source · 12 passages
Search aliases: n
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
n'Use in source · 1 passage
Search aliases: n'
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\operatorname{N}_{L/L^H}Use in source · 2 passages
Search aliases: N_L/L^H operatorname
Use in source · chapters04_06.tex, line 248
In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs a nonlinear transfer.
Exact mathematical passage
\operatorname{N}_{L/L^H}(x)=\prod_{h\in H}h(x). \label{eq:b-norm-product}
\operatorname{N}_{L/L^K}Use in source · 1 passage
Search aliases: N_L/L^K operatorname
Use in source · chapters04_06.tex, line 255
the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. This gives Tr_L/L^K =Tr_L^H/L^K∘Tr_L/L^H, N_L/L^K =N_L^H/L^K∘N_L/L^H. The norm is homogeneous of degree |H| over the fixed field. It is multiplicative…
Exact mathematical passage
\operatorname{Tr}_{L/L^K} =\operatorname{Tr}_{L^H/L^K}\circ\operatorname{Tr}_{L/L^H}, \qquad \operatorname{N}_{L/L^K} =\operatorname{N}_{L^H/L^K}\circ\operatorname{N}_{L/L^H}.
\operatorname{N}_{L/\Q}Use in source · 1 passage
Search aliases: N_L/Q operatorname Q
Use in source · chapters04_06.tex, line 267
[One projection, different norms] In L=Q(sqrt2), let H be the full group. For x=a+bsqrt2, P_H(x)=a, Tr_L/Q(x)=2a, N_L/Q(x)=a^2-2b^2. The elements 1 and 1+sqrt2 have the same projection and the same trace, but their norms are 1 and -1. There is therefore no function of the projected value alone that recovers the norm on all of L. Similarly, P_H(sqrt2)=0, while P_H((sqrt2)^2)=2, so the projection is not multiplicative.
Exact mathematical passage
P_H(x)=a, \qquad \operatorname{Tr}_{L/\Q}(x)=2a, \qquad \operatorname{N}_{L/\Q}(x)=a^2-2b^2.
\operatorname{N}_{L^H/L^K}Use in source · 1 passage
Search aliases: N_L^H/L^K operatorname
Use in source · chapters04_06.tex, line 255
the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. This gives Tr_L/L^K =Tr_L^H/L^K∘Tr_L/L^H, N_L/L^K =N_L^H/L^K∘N_L/L^H. The norm is homogeneous of degree |H| over the fixed field. It is multiplicative…
Exact mathematical passage
\operatorname{Tr}_{L/L^K} =\operatorname{Tr}_{L^H/L^K}\circ\operatorname{Tr}_{L/L^H}, \qquad \operatorname{N}_{L/L^K} =\operatorname{N}_{L^H/L^K}\circ\operatorname{N}_{L/L^H}.
oUse in source · 1 passage
Search aliases: o
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\overline\QUse in source · 3 passages
Search aliases: overlineQ overline Q
Use in source · chapters04_06.tex, line 387
Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a continuous homomorphism widetildeρ:G_Q⟶ E, π∘widetildeρ=ρ. It is a weak solution wh
Exact mathematical passage
\overline\Q
\overline{\Q}Use in source · 2 passages
Search aliases: overlineQ overline Q
Use in source · appendices.tex, line 12
An arithmetic embedding problem uses the profinite group G_Q=Gal(overlineQ/Q) with its Krull topology. A finite target group has the discrete topology. The lift is continuous, compatible with the prescribed quotient, and proper when surjective. The finite cyclic examples in Chapter~6 use a different ambient group and are explicitly models.
Exact mathematical passage
G_{\Q}=\Gal(\overline{\Q}/\Q)
\oversetUse in source · 1 passage
Search aliases: overset
Use in source · chapters04_06.tex, line 396
Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a continuous homomorphism widetildeρ:G_Q⟶ E, π∘widetildeρ=ρ. It is a weak solution when this equation holds, and a proper solution when widetildeρ is al…
Exact mathematical passage
1\longrightarrow N\longrightarrow E\overset{\pi}{\longrightarrow}G\longrightarrow1. \label{eq:b-group-extension}
PUse in source · 10 passages
Search aliases: P
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
P_H
pUse in source · 13 passages
Search aliases: p
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
P_aUse in source · 1 passage
Search aliases: P_a
Use in source · chapters01_03.tex, line 339
An orbit polynomial gives a useful piece of the theorem's mechanism. For any a∈ E, P_a(T)=∏_h∈ G(T-σ_h(a)) has coefficients in K, since the action permutes its factors. Thus every element of E is algebraic over K. Orbit repetitions are allowed, so this polynomial need not be the minimal polynomial and can have repeated factors. The fact that all such elements together generate a finite extension of degree exactly |G| is the content of Artin's theorem. Bounded degrees for individual elements shou…
Exact mathematical passage
P_a(T)=\prod_{h\in G}(T-\sigma_h(a))
P_HUse in source · 5 passages
Search aliases: P_H
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
P_H
\pmodUse in source · 4 passages
Search aliases: pmod
Use in source · chapters01_03.tex, line 261
Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreducible. Its derivative 3X^2+6X+2 is nonzero and has smaller degree, so it cannot have a common factor with this irreducible cubic. The cubic's value at t…
Exact mathematical passage
\begin{array}{c|rrrrrrr} x&0&1&2&3&4&5&6\\\hline x^3+3x^2+2x+5\pmod7&5&4&1&2&6&5&5. \end{array}
QUse in source · 3 passages
Search aliases: Q
Use in source · chapters04_06.tex, line 116
There is also an exact projector test. If P,Q are orthogonal projections and PQ is an orthogonal projection, then self adjointness gives PQ=(PQ)^*=QP. In the compatible case, E_HE_K is the projection onto the common fixed range. In the incompatible case, one must retain the ordered products as transfers. Replacing them by a shared refinement would remove a real commutator.
Exact mathematical passage
P,Q
\mathbb QUse in source · 1 passage
Search aliases: Q mathbb
Use in source · bibliography.tex, line 43
bibitemHuangEtAl2026 Xiaoyu Huang, Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries and Shaowu Zhang. The Mathieu group M_23 is a Galois group over Q. arXiv:2608.08538v2, submitted 7 October 2026. Manuscript dated 4 October 2026. Preprint used for dated context. https://arxiv.org/abs/2608.08538v2.
Exact mathematical passage
\mathbb Q
qUse in source · 2 passages
Search aliases: q
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\Q^4Use in source · 3 passages
Search aliases: Q^4 Q
Use in source · chapters04_06.tex, line 196
[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity.
Exact mathematical passage
\Q^4
\Q^GUse in source · 4 passages
Search aliases: Q^G Q
Use in source · chapters01_03.tex, line 73
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
\Q^G
\Q^\timesUse in source · 1 passage
Search aliases: Q^× Q times
Use in source · chapters01_03.tex, line 106
The polynomial already splits in L, so L/Q is Galois and Gal(L/Q)={1,σ}≃ C_2. Its fixed elements satisfy a+bα=a-bα, whence b=0. This directly checks L^⟨σ⟩=Q without appealing to an abstract correspondence. The same argument works for every nonsquare d∈Q^×, with sqrt2 replaced by sqrt d.
Exact mathematical passage
d\in\Q^\times
RUse in source · 1 passage
Search aliases: R
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\RUse in source · 1 passage
Search aliases: R
Use in source · chapters01_03.tex, line 222
The field Q(α) is contained in R, whereas ζ is nonreal. Thus ζ∉Q(α). Its quadratic equation remains irreducible over this real field, and [L:Q]=[L:Q(α)] [Q(α):Q]=2·3=6. The faithful action on three roots embeds Gal(L/Q) into S_3. Both groups have order six, so the embedding is an isomorphism.
Exact mathematical passage
\R
r^{-1}Use in source · 3 passages
Search aliases: r^-1
Use in source · chapters04_06.tex, line 120
[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, because a,b generate S_3. The rank one projector onto that intersection exists, even though it is not their ordered product.
Exact mathematical passage
ba=r^{-1}
r^2Use in source · 2 passages
Search aliases: r^2
Use in source · chapters01_03.tex, line 204
[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/05_cyclic_cubic.pdf captionThe cubic realization closes inside one root field. The transformation r↦ r^2-2 cycles its three roots, and their exact symmetric identities recover the original polynomial. The group has order three because the polynomial is irreducible and already splits in that field.
Exact mathematical passage
r\mapsto r^2-2
r^3Use in source · 1 passage
Search aliases: r^3
Use in source · chapters04_06.tex, line 204
But krk=(1342) is not among 1,r,r^2,r^3, so H is not normal. Moreover kr∈ KH does not lie in HK. If it equalled r^i, then k∈ H, which is false. If it equalled r^ik, then krk=r^i, which is also false. Hence HK≠ KH. Both reverse implications fail in this faithful group action. They hold in the regular action because its group algebra representation retains the missing information.
Exact mathematical passage
1,r,r^2,r^3
r^iUse in source · 3 passages
Search aliases: r^i
Use in source · chapters04_06.tex, line 204
But krk=(1342) is not among 1,r,r^2,r^3, so H is not normal. Moreover kr∈ KH does not lie in HK. If it equalled r^i, then k∈ H, which is false. If it equalled r^ik, then krk=r^i, which is also false. Hence HK≠ KH. Both reverse implications fail in this faithful group action. They hold in the regular action because its group algebra representation retains the missing information.
Exact mathematical passage
r^i
r_1Use in source · 3 passages
Search aliases: r_1
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
r_1,r_2,r_3
r_2Use in source · 1 passage
Search aliases: r_2
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
r_1,r_2,r_3
r_3Use in source · 1 passage
Search aliases: r_3
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
r_1,r_2,r_3
r_gUse in source · 8 passages
Search aliases: r_g
Use in source · chapters07_08.tex, line 16
For every g∈ G, specify a polynomial r_g∈Q[X] of degree less than n. It is intended to describe σ_g(θ)=r_g(θ). The equality f(r_g(θ))=0 ensures that this substitution induces a unital Q-algebra endomorphism of L. Because L is a field, the endomorphism is injective. Because it is a linear endomorphism of a finite dimensional vector space, it is surjective. Thus it is an automorphism.
Exact mathematical passage
r_g\in\Q[X]
r_{gh}Use in source · 1 passage
Search aliases: r_gh
Use in source · chapters07_08.tex, line 23
The group law must still hold. With our convention that σ_gσ_h means apply σ_h first, the required congruences are r_1(X) · ≡ Xpmod f, r_h(r_g(X)) · ≡ r_gh(X)pmod f. The reversed order of the displayed polynomial substitutions is deliberate. Applying σ_g to r_h(θ) replaces θ by r_g(θ), since rational coefficients are fixed. Finally the residues r_gbmod f must be pairwise distinct. Equal generator images would give equal automorphisms and could collapse the intended group.
Exact mathematical passage
r_1(X)&\equiv X\pmod f,\label{eq:g07-unit}\\ r_h(r_g(X))&\equiv r_{gh}(X)\pmod f.\label{eq:g07-composition}
r_hUse in source · 2 passages
Search aliases: r_h
Use in source · chapters07_08.tex, line 23
The group law must still hold. With our convention that σ_gσ_h means apply σ_h first, the required congruences are r_1(X) · ≡ Xpmod f, r_h(r_g(X)) · ≡ r_gh(X)pmod f. The reversed order of the displayed polynomial substitutions is deliberate. Applying σ_g to r_h(θ) replaces θ by r_g(θ), since rational coefficients are fixed. Finally the residues r_gbmod f must be pairwise distinct. Equal generator images would give equal automorphisms and could collapse the intended group.
Exact mathematical passage
r_1(X)&\equiv X\pmod f,\label{eq:g07-unit}\\ r_h(r_g(X))&\equiv r_{gh}(X)\pmod f.\label{eq:g07-composition}
r_iUse in source · 1 passage
Search aliases: r_i
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
V=\prod_{i<j}(r_i-r_j)
r_jUse in source · 1 passage
Search aliases: r_j
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
V=\prod_{i<j}(r_i-r_j)
r_sUse in source · 3 passages
Search aliases: r_s
Use in source · chapters07_08.tex, line 58
For G=C_2={1,s}, take f=X^2-2 and choose r_1=X, r_s=-X. The polynomial is irreducible over Q, because two is not a rational square. Both residues satisfy f(r_g)≡0pmod f. The nontrivial composition gives r_s(r_s(X))=X, and the two images are distinct. The certificate theorem returns Q(sqrt2)/Q with its full group C_2.
Exact mathematical passage
r_1=X,\qquad r_s=-X.
\operatorname{ran}Use in source · 1 passage
Search aliases: ran operatorname
Use in source · appendices.tex, line 10
Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.
Exact mathematical passage
\operatorname{ran}(E_K)\subseteq\operatorname{ran}(E_H)
SUse in source · 32 passages
Search aliases: S
Use in source · appendices.tex, line 10
Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.
Exact mathematical passage
S_4
sUse in source · 42 passages
Search aliases: s
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
s^2Use in source · 1 passage
Search aliases: s^2
Use in source · chapters01_03.tex, line 121
The four elements 1, sqrt2, sqrt3, sqrt6 form a rational basis. Independent choices of signs define four automorphisms. Let s change the sign of sqrt2 and let t change the sign of sqrt3. They satisfy s^2=t^2=1, st=ts. They are distinct because their effects on the two generators are distinct. The extension is the splitting field of (X^2-2)(X^2-3), so it is Galois. Since the degree is four, these four automorphisms are the full group: Gal(L/Q)≃ C_2× C_2=V_4.
Exact mathematical passage
s^2=t^2=1,\qquad st=ts.
S_3Use in source · 19 passages
Search aliases: S_3
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
S_4Use in source · 13 passages
Search aliases: S_4
Use in source · appendices.tex, line 10
Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.
Exact mathematical passage
S_4
S_dUse in source · 1 passage
Search aliases: S_d
Use in source · chapters01_03.tex, line 52
Suppose f has d distinct roots in its splitting field L. Every automorphism of L/Q permutes them. The action is faithful because the roots generate L: an automorphism fixing every root fixes every rational expression in those roots. Consequently Gal(L/Q)↪ S_d. If f is irreducible, the action is transitive. One way to see this is to send any root to any other through the isomorphism between the corresponding simple extensions, then extend that embedding to the splitting field. Normality ensures t…
Exact mathematical passage
\label{eq:g01-root-action} \Gal(L/\Q)\hookrightarrow S_d.
\sectionUse in source · 1 passage
Search aliases: section
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
T^2Use in source · 3 passages
Search aliases: T^2
Use in source · chapters01_03.tex, line 415
The smallest example makes the warning exact. Let G=C_2 exchange two variables x,y. Define s=x+y, p=xy. The variables x,y are the roots of T^2-sT+p. Since y=s-x, the field Q(x,y) has degree at most two over Q(s,p). The nontrivial exchange fixes s,p, and x∉Q(s,p) because an element in that field would be fixed. Hence the degree is exactly two. Artin's theorem then shows Q(x,y)^C_2=Q(s,p). The parameters s,p are algebraically independent: the algebraic extension to Q(x,y) preserves transcendence d…
Exact mathematical passage
T^2-sT+p
t^2Use in source · 1 passage
Search aliases: t^2
Use in source · chapters01_03.tex, line 121
The four elements 1, sqrt2, sqrt3, sqrt6 form a rational basis. Independent choices of signs define four automorphisms. Let s change the sign of sqrt2 and let t change the sign of sqrt3. They satisfy s^2=t^2=1, st=ts. They are distinct because their effects on the two generators are distinct. The extension is the splitting field of (X^2-2)(X^2-3), so it is Galois. Since the degree is four, these four automorphisms are the full group: Gal(L/Q)≃ C_2× C_2=V_4.
Exact mathematical passage
s^2=t^2=1,\qquad st=ts.
\operatorname{Tr}Use in source · 3 passages
Search aliases: Tr operatorname
Use in source · chapters04_06.tex, line 242
In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs a nonlinear transfer.
Exact mathematical passage
P_H(x)=\frac1{|H|}\sum_{h\in H}h(x) =\frac1{|H|}\operatorname{Tr}_{L/L^H}(x). \label{eq:b-trace-average}
\operatorname{Tr}_{L/L^H}Use in source · 2 passages
Search aliases: Tr_L/L^H operatorname
Use in source · chapters04_06.tex, line 242
In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs a nonlinear transfer.
Exact mathematical passage
P_H(x)=\frac1{|H|}\sum_{h\in H}h(x) =\frac1{|H|}\operatorname{Tr}_{L/L^H}(x). \label{eq:b-trace-average}
\operatorname{Tr}_{L/L^K}Use in source · 1 passage
Search aliases: Tr_L/L^K operatorname
Use in source · chapters04_06.tex, line 255
the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. This gives Tr_L/L^K =Tr_L^H/L^K∘Tr_L/L^H, N_L/L^K =N_L^H/L^K∘N_L/L^H. The norm is homogeneous of degree |H| over the fixed field. It is multiplicative…
Exact mathematical passage
\operatorname{Tr}_{L/L^K} =\operatorname{Tr}_{L^H/L^K}\circ\operatorname{Tr}_{L/L^H}, \qquad \operatorname{N}_{L/L^K} =\operatorname{N}_{L^H/L^K}\circ\operatorname{N}_{L/L^H}.
\operatorname{Tr}_{L/\Q}Use in source · 1 passage
Search aliases: Tr_L/Q operatorname Q
Use in source · chapters04_06.tex, line 267
[One projection, different norms] In L=Q(sqrt2), let H be the full group. For x=a+bsqrt2, P_H(x)=a, Tr_L/Q(x)=2a, N_L/Q(x)=a^2-2b^2. The elements 1 and 1+sqrt2 have the same projection and the same trace, but their norms are 1 and -1. There is therefore no function of the projected value alone that recovers the norm on all of L. Similarly, P_H(sqrt2)=0, while P_H((sqrt2)^2)=2, so the projection is not multiplicative.
Exact mathematical passage
P_H(x)=a, \qquad \operatorname{Tr}_{L/\Q}(x)=2a, \qquad \operatorname{N}_{L/\Q}(x)=a^2-2b^2.
\operatorname{Tr}_{L^H/L^K}Use in source · 1 passage
Search aliases: Tr_L^H/L^K operatorname
Use in source · chapters04_06.tex, line 255
the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. This gives Tr_L/L^K =Tr_L^H/L^K∘Tr_L/L^H, N_L/L^K =N_L^H/L^K∘N_L/L^H. The norm is homogeneous of degree |H| over the fixed field. It is multiplicative…
Exact mathematical passage
\operatorname{Tr}_{L/L^K} =\operatorname{Tr}_{L^H/L^K}\circ\operatorname{Tr}_{L/L^H}, \qquad \operatorname{N}_{L/L^K} =\operatorname{N}_{L^H/L^K}\circ\operatorname{N}_{L/L^H}.
\operatorname{trdeg}Use in source · 1 passage
Search aliases: trdeg operatorname
Use in source · chapters01_03.tex, line 364
More generally, E/K is algebraic, so the two fields have the same transcendence degree over Q: trdeg_QK=trdeg_QE=|G|. The finite degree [E:K] does not make E or K finite over Q. Both retain transcendental parameters. For the trivial group, the construction says only that E=K=Q(x_1), whereas its arithmetic realization was simply Q. Even this endpoint shows why changing the base cannot be silently ignored.
Exact mathematical passage
\label{eq:g03-transdegree} \operatorname{trdeg}_{\Q}K=\operatorname{trdeg}_{\Q}E=|G|.
\operatorname{trdeg}_{\Q}Use in source · 1 passage
Search aliases: trdeg_Q operatorname Q
Use in source · chapters01_03.tex, line 364
More generally, E/K is algebraic, so the two fields have the same transcendence degree over Q: trdeg_QK=trdeg_QE=|G|. The finite degree [E:K] does not make E or K finite over Q. Both retain transcendental parameters. For the trivial group, the construction says only that E=K=Q(x_1), whereas its arithmetic realization was simply Q. Even this endpoint shows why changing the base cannot be silently ignored.
Exact mathematical passage
\label{eq:g03-transdegree} \operatorname{trdeg}_{\Q}K=\operatorname{trdeg}_{\Q}E=|G|.
UUse in source · 1 passage
Search aliases: U
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
uUse in source · 22 passages
Search aliases: u
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
u^3Use in source · 1 passage
Search aliases: u^3
Use in source · chapters04_06.tex, line 357
The generic splitting field also has group S_3. To prove this without an experimental inference, take a transcendental u and put t=u^3. The polynomial X^3-t is Eisenstein at the prime t in Q[t][X], hence irreducible over Q(t). Its root field is Q(u), and the splitting field is Q(u,ζ). The only elements of Q(u) algebraic over Q lie in Q. Indeed, if a nonconstant rational function v(u) were algebraic over Q, clearing the equation defining v would make u algebraic over Q(v), contradicting its trans…
Exact mathematical passage
t=u^3
vUse in source · 8 passages
Search aliases: v
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
V^2Use in source · 2 passages
Search aliases: V^2
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
V^2=49
V^HUse in source · 6 passages
Search aliases: V^H
Use in source · chapters04_06.tex, line 35
[Fixed range and rank] The operator E_H is the orthogonal projector onto V^H={v∈ V:λ(h)v=v for every h∈ H}. Its rank is [G:H]. If H≤ K, then E_HE_K=E_KE_H=E_K, V^K⊆ V^H.
Exact mathematical passage
V^H=\{v\in V:\lambda(h)v=v\text{ for every }h\in H\}.
V^KUse in source · 2 passages
Search aliases: V^K
Use in source · chapters04_06.tex, line 39
[Fixed range and rank] The operator E_H is the orthogonal projector onto V^H={v∈ V:λ(h)v=v for every h∈ H}. Its rank is [G:H]. If H≤ K, then E_HE_K=E_KE_H=E_K, V^K⊆ V^H.
Exact mathematical passage
E_HE_K=E_KE_H=E_K, \qquad V^K\subseteq V^H. \label{eq:b-nested-averages}
V^{\langle H,K\rangle}Use in source · 1 passage
Search aliases: V^⟨ H,K⟩ langle rangle
Use in source · chapters04_06.tex, line 104
rion] In the regular realization, the following statements are equivalent: HK=KH, HK is a subgroup, E_HE_K=E_KE_H. When they hold, E_HE_K=E_KE_H=E_HK=E_⟨ H,K⟩. Without a compatibility assumption, the intersection of the fixed ranges is still V^H∩ V^K=V^⟨ H,K⟩.
Exact mathematical passage
V^H\cap V^K=V^{\langle H,K\rangle}. \label{eq:b-fixed-intersection}
V_4Use in source · 4 passages
Search aliases: V_4
Use in source · chapters01_03.tex, line 108
Two independent signs: V_4.
Exact mathematical passage
V_4
v_{Hg}Use in source · 1 passage
Search aliases: v_Hg
Use in source · chapters04_06.tex, line 55
The orbits of H on the basis labels are the right cosets Hg. For one representative g from each coset, put v_Hg=|H|^-1/2∑_h∈ Hδ_hg. These vectors have disjoint supports, are orthonormal, and span the fixed range. Their number is [G:H]. Finally H≤ K makes every K fixed vector H fixed. Orthogonal projectors onto nested subspaces multiply to the projector onto the smaller subspace, in either order. This proves [source reference].
Exact mathematical passage
v_{Hg}=|H|^{-1/2}\sum_{h\in H}\delta_{hg}.
WUse in source · 3 passages
Search aliases: W
Use in source · chapters04_06.tex, line 197
[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity.
Exact mathematical passage
W=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\}.
wUse in source · 1 passage
Search aliases: w
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\widetilde\rhoUse in source · 18 passages
Search aliases: widetildeρ widetilde rho
Use in source · chapters04_06.tex, line 401
tinuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a continuous homomorphism widetildeρ:G_Q⟶ E, π∘widetildeρ=ρ. It is a weak solution when this equation holds, and a proper solution when widetildeρ is also surjective.
Exact mathematical passage
\widetilde\rho:G_{\Q}\longrightarrow E, \qquad \pi\circ\widetilde\rho=\rho. \label{eq:b-lift-equation}
\widetilde\rho_0Use in source · 1 passage
Search aliases: widetildeρ_0 widetilde rho
Use in source · chapters04_06.tex, line 453
[The kernel is missed in a direct product] Take E=N× G with π(n,g)=g. The section s(g)=(1,g) gives widetildeρ_0(γ)=(1,ρ(γ)). Its image is {1}× G. It reaches every element of the desired quotient, but reaches none of the nonidentity elements of N×{1}. With the finite toy N=G=Γ=C_2 and ρ the identity, the image has two elements while E has four. All lift equations hold, and properness fails.
Exact mathematical passage
\widetilde\rho_0(\gamma)=(1,\rho(\gamma)).
X^2Use in source · 12 passages
Search aliases: X^2
Use in source · chapters01_03.tex, line 94
Let α=sqrt2 and L=Q(α). The polynomial X^2-2 is irreducible over Q. Indeed, if a/b in lowest terms had square two, then a^2=2b^2 would make a even. Writing a=2c gives b^2=2c^2, so b would also be even, a contradiction.
Exact mathematical passage
X^2-2
x^2Use in source · 1 passage
Search aliases: x^2
Use in source · chapters01_03.tex, line 261
Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreducible. Its derivative 3X^2+6X+2 is nonzero and has smaller degree, so it cannot have a common factor with this irreducible cubic. The cubic's value at t…
Exact mathematical passage
\begin{array}{c|rrrrrrr} x&0&1&2&3&4&5&6\\\hline x^3+3x^2+2x+5\pmod7&5&4&1&2&6&5&5. \end{array}
x^3Use in source · 1 passage
Search aliases: x^3
Use in source · chapters01_03.tex, line 261
Modulo two, irreducibility gives a four cycle. Squarefreeness holds since the derivative of X^4+X+1 is one. Modulo seven there is the exact factorization f(X)=(X-3)(X^3+3X^2+2X+5) in F_7[X]. The cubic takes the following values: c|rrrrrrr x · 0 · 1 · 2 · 3 · 4 · 5 · 6 x^3+3x^2+2x+5pmod7 · 5 · 4 · 1 · 2 · 6 · 5 · 5. It has no root, hence is irreducible. Its derivative 3X^2+6X+2 is nonzero and has smaller degree, so it cannot have a common factor with this irreducible cubic. The cubic's value at t…
Exact mathematical passage
\begin{array}{c|rrrrrrr} x&0&1&2&3&4&5&6\\\hline x^3+3x^2+2x+5\pmod7&5&4&1&2&6&5&5. \end{array}
X^4Use in source · 6 passages
Search aliases: X^4
Use in source · chapters01_03.tex, line 240
For a quartic realization, consider f(X)=X^4-X-1. Its reduction modulo two is X^4+X+1. It has no root in F_2. The only monic irreducible quadratic over F_2 is X^2+X+1. Modulo that polynomial, X^2=X+1, X^3=1 and X^4=X, so X^4+X+1 has remainder one. A reducible quartic without a linear factor must factor as two irreducible quadratics. This possibility has just been excluded, proving that the reduction is irreducible. Gauss's lemma now proves that f is irreducible over Q.
Exact mathematical passage
f(X)=X^4-X-1.
x_1Use in source · 10 passages
Search aliases: x_1
Use in source · chapters01_03.tex, line 300
equation σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\sigma_h(x_1)=x_h
x_2Use in source · 1 passage
Search aliases: x_2
Use in source · chapters04_06.tex, line 197
[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity.
Exact mathematical passage
W=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\}.
x_3Use in source · 1 passage
Search aliases: x_3
Use in source · chapters04_06.tex, line 197
[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity.
Exact mathematical passage
W=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\}.
x_4Use in source · 1 passage
Search aliases: x_4
Use in source · chapters04_06.tex, line 197
[A faithful group action that hides the boundary] Let S_4 act by permuting the coordinates of Q^4, and restrict to W={(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0}. This standard representation is faithful. Indeed, a permutation acting trivially on W also acts trivially on the constant line. Their direct sum is Q^4, so its permutation matrix is the identity.
Exact mathematical passage
W=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\}.
x_hUse in source · 1 passage
Search aliases: x_h
Use in source · chapters01_03.tex, line 300
equation σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\sigma_h(x_1)=x_h
x_{hg}Use in source · 1 passage
Search aliases: x_hg
Use in source · chapters01_03.tex, line 296
For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\label{eq:g03-variable-action} \sigma_h(x_g)=x_{hg},\qquad\sigma_h(q)=q\quad(q\in\Q).
yUse in source · 13 passages
Search aliases: y
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
\ZUse in source · 3 passages
Search aliases: Z
Use in source · chapters01_03.tex, line 47
An integral polynomial can be used if desired. Clear denominators to obtain a polynomial in Z[X] with the same roots. The resulting polynomial need not be monic. Alternatively, multiply a primitive element by an integer large enough to make it integral and use its monic minimal polynomial. These changes concern presentation. They do not replace the requirement that the splitting field have the correct group.
Exact mathematical passage
\Z[X]
zUse in source · 2 passages
Search aliases: z
Use in source · appendices.tex, line 18
[4pt] LC04 · Every finite solvable group occurs over Q, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement. [4pt] LC08 · Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial. [4pt] LC09 · Irreducible cubic discriminant classification over Q. The characteristic restrictions in the source are recorded explicitly. [4pt] LC10 · Artin's finite automorphism theorem for an a…
Exact mathematical passage
4pt] LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt] LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt] LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt] LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt] LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt] LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt] LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt] LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt] LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt] LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt] LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\ \bottomrule \end{longtable} The full contracts include exact source versions, printed page numbers, notation translations and acquisition diagnostics. They are under \texttt{literature/} in the source bundle. The bibliography records primary source URLs. Full third-party copies are omitted from the distributed bundle. \section{The finite computation actually executed} The script \texttt{computations/exact\_examples.py} uses exact rational, permutation and finite-field arithmetic. It records five scalar polynomials and their discriminants. It verifies the quartic's reductions at two and seven. It constructs all six subgroups of $S_3$ and checks all $36$ ordered pairs, including each subgroup paired with itself. It also records the faithful standard $S_4$ representation example, two finite lifting models and the three specialization values zero, one and two. For the ordered basis \[ (),\ (23),\ (12),\ (123),\ (132),\ (13)
|^{-1}Use in source · 1 passage
Search aliases: |^-1
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
E_H=|H|^{-1}\sum_{h\in H}\lambda(h)
|^{-1/2}Use in source · 1 passage
Search aliases: |^-1/2
Use in source · chapters04_06.tex, line 55
The orbits of H on the basis labels are the right cosets Hg. For one representative g from each coset, put v_Hg=|H|^-1/2∑_h∈ Hδ_hg. These vectors have disjoint supports, are orthonormal, and span the fixed range. Their number is [G:H]. Finally H≤ K makes every K fixed vector H fixed. Orthogonal projectors onto nested subspaces multiply to the projector onto the smaller subspace, in either order. This proves [source reference].
Exact mathematical passage
v_{Hg}=|H|^{-1/2}\sum_{h\in H}\delta_{hg}.
|^2Use in source · 1 passage
Search aliases: |^2
Use in source · chapters04_06.tex, line 48
Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed vector is unchanged by every summand, and therefore by the average.
Exact mathematical passage
E_H^2=\frac1{|H|^2}\sum_{h,k\in H}\lambda(hk) =\frac1{|H|}\sum_{u\in H}\lambda(u)=E_H.
\cdotUse in source · 4 passages
Search aliases: · cdot
Use in source · chapters01_03.tex, line 70
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
g\cdot h=gh.
\timesUse in source · 5 passages
Search aliases: × times
Use in source · chapters01_03.tex, line 106
The polynomial already splits in L, so L/Q is Galois and Gal(L/Q)={1,σ}≃ C_2. Its fixed elements satisfy a+bα=a-bα, whence b=0. This directly checks L^⟨σ⟩=Q without appealing to an abstract correspondence. The same argument works for every nonsquare d∈Q^×, with sqrt2 replaced by sqrt d.
Exact mathematical passage
d\in\Q^\times
\alpha^2Use in source · 4 passages
Search aliases: α^2 alpha
Use in source · chapters01_03.tex, line 100
Every element of L has a unique expression a+bα with a,b∈Q. The map σ(a+bα)=a-bα preserves addition and multiplication because both α and -α satisfy the same relation α^2=2. It fixes Q, has square the identity, and is nontrivial. The only possible images of α under a rational automorphism are α and -α. Thus there are exactly two automorphisms.
Exact mathematical passage
\alpha^2=2
\alpha^3Use in source · 1 passage
Search aliases: α^3 alpha
Use in source · chapters01_03.tex, line 160
We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from α(β+γ)=-α-α^2, βγ=α^2+α-2. The third follows by multiplying the latter expression by α and using the cubic relation. These are exact algebraic equalit…
Exact mathematical passage
\alpha^3=-\alpha^2+2\alpha+1, \qquad \alpha^4=3\alpha^2-\alpha-1.
\alpha^4Use in source · 1 passage
Search aliases: α^4 alpha
Use in source · chapters01_03.tex, line 160
We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from α(β+γ)=-α-α^2, βγ=α^2+α-2. The third follows by multiplying the latter expression by α and using the cubic relation. These are exact algebraic equalit…
Exact mathematical passage
\alpha^3=-\alpha^2+2\alpha+1, \qquad \alpha^4=3\alpha^2-\alpha-1.
\betaUse in source · 7 passages
Search aliases: β beta
Use in source · chapters01_03.tex, line 166
We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from α(β+γ)=-α-α^2, βγ=α^2+α-2. The third follows by multiplying the latter expression by α and using the cubic relation. These are exact algebraic equalit…
Exact mathematical passage
\beta=\alpha^2-2, \qquad \gamma=-\alpha^2-\alpha+1.
\beta^2Use in source · 1 passage
Search aliases: β^2 beta
Use in source · chapters01_03.tex, line 194
Since it splits there, the root field is already its splitting field. Its Galois group has order three and is therefore cyclic. There is an explicit generator sending α to β. The map is well defined because β is another root of the same irreducible polynomial. It is a field embedding into Q(α) and hence an automorphism, since the image has degree three. The identities β^2-2=γ, γ^2-2=α show that this automorphism cycles the three roots. Their distinctness also follows from the nonzero discriminan…
Exact mathematical passage
\beta^2-2=\gamma, \qquad \gamma^2-2=\alpha
\GammaUse in source · 11 passages
Search aliases: Γ Gamma
Use in source · chapters04_06.tex, line 432
[The identity of C_2 does not lift to C_4] Use additive notation and take Γ=Z/2Z, E=Z/4Z, G=Z/2Z, π(x)=xbmod2, ρ=id_Γ. If widetildeρ:Γ→ E were a lift, the image y of the nonzero element would satisfy π(y)=1, so y would be 1 or 3 modulo four. But the relation 1+1=0 in Γ forces 2y=0 in E, which holds only for y=0 or 2. No such lift exists.
Exact mathematical passage
\Gamma=\Z/2\Z, \quad E=\Z/4\Z, \quad G=\Z/2\Z, \quad \pi(x)=x\bmod2, \quad \rho=\operatorname{id}_{\Gamma}.
\gammaUse in source · 16 passages
Search aliases: γ gamma
Use in source · chapters01_03.tex, line 166
We can see the splitting field without using the discriminant criterion. Let α be a root. In Q(α), the defining equation gives α^3=-α^2+2α+1, α^4=3α^2-α-1. Set β=α^2-2, γ=-α^2-α+1. Direct reduction by the two displayed identities gives α+β+γ=-1, αβ+αγ+βγ=-2, αβγ=1. Thus f(X)=(X-α)(X-β)(X-γ) in Q(α)[X]. For example, the second identity follows from α(β+γ)=-α-α^2, βγ=α^2+α-2. The third follows by multiplying the latter expression by α and using the cubic relation. These are exact algebraic equalit…
Exact mathematical passage
\beta=\alpha^2-2, \qquad \gamma=-\alpha^2-\alpha+1.
\gamma^2Use in source · 1 passage
Search aliases: γ^2 gamma
Use in source · chapters01_03.tex, line 194
Since it splits there, the root field is already its splitting field. Its Galois group has order three and is therefore cyclic. There is an explicit generator sending α to β. The map is well defined because β is another root of the same irreducible polynomial. It is a field embedding into Q(α) and hence an automorphism, since the image has degree three. The identities β^2-2=γ, γ^2-2=α show that this automorphism cycles the three roots. Their distinctness also follows from the nonzero discriminan…
Exact mathematical passage
\beta^2-2=\gamma, \qquad \gamma^2-2=\alpha
\DeltaUse in source · 2 passages
Search aliases: Δ Delta
Use in source · chapters01_03.tex, line 208
. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discriminant itself by Δ, so V^2=Δ.
Exact mathematical passage
\Delta
\delta_aUse in source · 1 passage
Search aliases: δ_a delta
Use in source · chapters04_06.tex, line 121
[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, because a,b generate S_3. The rank one projector onto that intersection exists, even though it is not their ordered product.
Exact mathematical passage
E_HE_K\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_r), \qquad E_KE_H\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_{r^{-1}}).
\delta_bUse in source · 1 passage
Search aliases: δ_b delta
Use in source · chapters04_06.tex, line 121
[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, because a,b generate S_3. The rank one projector onto that intersection exists, even though it is not their ordered product.
Exact mathematical passage
E_HE_K\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_r), \qquad E_KE_H\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_{r^{-1}}).
\delta_{hg}Use in source · 1 passage
Search aliases: δ_hg delta
Use in source · chapters04_06.tex, line 55
The orbits of H on the basis labels are the right cosets Hg. For one representative g from each coset, put v_Hg=|H|^-1/2∑_h∈ Hδ_hg. These vectors have disjoint supports, are orthonormal, and span the fixed range. Their number is [G:H]. Finally H≤ K makes every K fixed vector H fixed. Orthogonal projectors onto nested subspaces multiply to the projector onto the smaller subspace, in either order. This proves [source reference].
Exact mathematical passage
v_{Hg}=|H|^{-1/2}\sum_{h\in H}\delta_{hg}.
\delta_rUse in source · 2 passages
Search aliases: δ_r delta
Use in source · chapters04_06.tex, line 121
[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, because a,b generate S_3. The rank one projector onto that intersection exists, even though it is not their ordered product.
Exact mathematical passage
E_HE_K\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_r), \qquad E_KE_H\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_{r^{-1}}).
\delta_{r^{-1}}Use in source · 2 passages
Search aliases: δ_r^-1 delta
Use in source · chapters04_06.tex, line 121
[Two transpositions in S_3] Let a=(12), b=(23), r=(123), and use permutation composition from right to left. Put H={1,a}, K={1,b}. Then ab=r, ba=r^-1, so E_HE_Kδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r), E_KE_Hδ_1=tfrac14(δ_1+δ_a+δ_b+δ_r^-1). In particular, [E_H,E_K]δ_1 =tfrac14(δ_r-δ_r^-1)≠0. Each average has rank three. Their common fixed range has rank one, because a,b generate S_3. The rank one projector onto that intersection exists, even though it is not their ordered product.
Exact mathematical passage
E_HE_K\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_r), \qquad E_KE_H\delta_1=\tfrac14(\delta_1+\delta_a+\delta_b+\delta_{r^{-1}}).
\zeta_3Use in source · 2 passages
Search aliases: ζ_3 zeta
Use in source · chapters04_06.tex, line 364
[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/17_parameter_bad.pdf captionThe specialization toy X^3-t has group S_3 at t=2, group C_2 at the nonsingular value t=1, and trivial splitting field at t=0. Its generic splitting field contains ζ_3, so this is not a regular family over Q.
Exact mathematical passage
\zeta_3
\etaUse in source · 2 passages
Search aliases: η eta
Use in source · chapters04_06.tex, line 474
For a direct product, a candidate lift has the form (η,ρ), where η:Γ→ N is a homomorphism. Even surjectivity of both coordinates does not automatically imply surjectivity of the pair. If Γ=N=G=C_2 and both maps are the identity, their combined image is the diagonal subgroup of order two. Proposition~[source reference] locates the failure: the kernel of ρ is trivial and supplies no nontrivial kernel element.
Exact mathematical passage
(\eta,\rho)
\thetaUse in source · 6 passages
Search aliases: θ theta
Use in source · chapters07_08.tex, line 11
Let G be a finite group of order n. A polynomial field presentation consists of a monic irreducible polynomial f∈Q[X] of degree n and L=Q[X]/(f), θ=Xbmod f. Irreducibility is the field input. The quotient is automatically a commutative algebra of dimension n, but that dimension alone would also hold for a reducible polynomial. A reducible quotient can have zero divisors or nilpotents and cannot serve as the requested field.
Exact mathematical passage
L=\Q[X]/(f),\qquad \theta=X\bmod f.
\muUse in source · 2 passages
Search aliases: μ mu
Use in source · chapters07_08.tex, line 75
The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before producing the basis. It cannot be used backwards to infer that an arbitrary choice of these coefficients has all the required properties.
Exact mathematical passage
\mu:\Q[G]\otimes_\Q\Q[G]\longrightarrow\Q[G], \qquad \mu(u,v)=\Phi^{-1}(\Phi(u)\Phi(v)).
\piUse in source · 12 passages
Search aliases: π pi
Use in source · chapters04_06.tex, line 396
Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a continuous homomorphism widetildeρ:G_Q⟶ E, π∘widetildeρ=ρ. It is a weak solution when this equation holds, and a proper solution when widetildeρ is al…
Exact mathematical passage
1\longrightarrow N\longrightarrow E\overset{\pi}{\longrightarrow}G\longrightarrow1. \label{eq:b-group-extension}
\rhoUse in source · 32 passages
Search aliases: ρ rho
Use in source · chapters01_03.tex, line 74
Let G be any finite group. Its left regular action is the permutation action on its own underlying set, g· h=gh. It is faithful: if g fixes the identity element, then g=1. On the vector space Q^G with basis e_h, the corresponding operators are ρ(g)e_h=e_gh. They satisfy ρ(g_1g_2)=ρ(g_1)ρ(g_2) and give an embedding G↪GL(Q^G). These are exact constructions for every finite group.
Exact mathematical passage
\rho(g)e_h=e_{gh}.
\SigmaUse in source · 8 passages
Search aliases: Σ Sigma
Use in source · chapters07_08.tex, line 38
The substitutions give a homomorphism from G into Aut_Q(L). The identity and composition congruences prove the homomorphism law. Distinct generator images make the homomorphism injective. Write its image as Σ, so |Σ|=n.
Exact mathematical passage
\Sigma
\sigmaUse in source · 13 passages
Search aliases: σ sigma
Use in source · chapters01_03.tex, line 97
Every element of L has a unique expression a+bα with a,b∈Q. The map σ(a+bα)=a-bα preserves addition and multiplication because both α and -α satisfy the same relation α^2=2. It fixes Q, has square the identity, and is nontrivial. The only possible images of α under a rational automorphism are α and -α. Thus there are exactly two automorphisms.
Exact mathematical passage
\sigma(a+b\alpha)=a-b\alpha
\sigma_gUse in source · 3 passages
Search aliases: σ_g sigma
Use in source · chapters07_08.tex, line 16
For every g∈ G, specify a polynomial r_g∈Q[X] of degree less than n. It is intended to describe σ_g(θ)=r_g(θ). The equality f(r_g(θ))=0 ensures that this substitution induces a unital Q-algebra endomorphism of L. Because L is a field, the endomorphism is injective. Because it is a linear endomorphism of a finite dimensional vector space, it is surjective. Thus it is an automorphism.
Exact mathematical passage
\sigma_g(\theta)=r_g(\theta)
\sigma_hUse in source · 7 passages
Search aliases: σ_h sigma
Use in source · chapters01_03.tex, line 296
For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\label{eq:g03-variable-action} \sigma_h(x_g)=x_{hg},\qquad\sigma_h(q)=q\quad(q\in\Q).
\sigma_{h^{-1}}Use in source · 1 passage
Search aliases: σ_h^-1 sigma
Use in source · chapters01_03.tex, line 300
For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\sigma_{h^{-1}}
\sigma_{hk}Use in source · 1 passage
Search aliases: σ_hk sigma
Use in source · chapters01_03.tex, line 300
For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\sigma_h\sigma_k=\sigma_{hk}
\sigma_kUse in source · 1 passage
Search aliases: σ_k sigma
Use in source · chapters01_03.tex, line 300
For h∈ G, define σ_h(x_g)=x_hg, σ_h(q)=q (q∈Q). A permutation of independent variables extends to an automorphism of the polynomial ring and then of its fraction field. The inverse is σ_h^-1. The action satisfies σ_hσ_k=σ_hk, so it is a group action by genuine field automorphisms. It is faithful because σ_h(x_1)=x_h differs from x_1 when h≠1.
Exact mathematical passage
\sigma_h\sigma_k=\sigma_{hk}
\tauUse in source · 2 passages
Search aliases: τ tau
Use in source · chapters01_03.tex, line 229
There is also a concrete pair of generators. Over Q(ζ), the element α has degree three: the tower degree is [L:Q(ζ)]=3. Sending α to ζα while fixing ζ therefore gives an automorphism τ of order three. Complex conjugation gives an automorphism c of order two that fixes α and sends ζ to ζ^2. On the generators, cτ c=τ^-1. Hence the cyclic cubic subgroup is inverted by the other generator, as in the usual presentation of S_3.
Exact mathematical passage
\tau
\tau^{-1}Use in source · 1 passage
Search aliases: τ^-1 tau
Use in source · chapters01_03.tex, line 230
There is also a concrete pair of generators. Over Q(ζ), the element α has degree three: the tower degree is [L:Q(ζ)]=3. Sending α to ζα while fixing ζ therefore gives an automorphism τ of order three. Complex conjugation gives an automorphism c of order two that fixes α and sends ζ to ζ^2. On the generators, cτ c=τ^-1. Hence the cyclic cubic subgroup is inverted by the other generator, as in the usual presentation of S_3.
Exact mathematical passage
c\tau c=\tau^{-1}.
\PhiUse in source · 2 passages
Search aliases: Φ Phi
Use in source · chapters07_08.tex, line 68
The normal basis theorem supplies a regular module for an already existing finite Galois extension [source reference]. If L/Q has group G and normal basis generator α, the vector-space map Φ:Q[G]⟶ L, ∑_g a_g g⟼∑_g a_g g(α) is an equivariant linear isomorphism. It transfers subgroup averages and their fixed vector spaces. It is not a claim that the usual multiplication in Q[G] is field multiplication.
Exact mathematical passage
\Phi:\Q[G]\longrightarrow L, \qquad \sum_g a_g g\longmapsto\sum_g a_g g(\alpha)
\Phi^{-1}Use in source · 1 passage
Search aliases: Φ^-1 Phi
Use in source · chapters07_08.tex, line 75
The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before producing the basis. It cannot be used backwards to infer that an arbitrary choice of these coefficients has all the required properties.
Exact mathematical passage
\mu:\Q[G]\otimes_\Q\Q[G]\longrightarrow\Q[G], \qquad \mu(u,v)=\Phi^{-1}(\Phi(u)\Phi(v)).
\toUse in source · 12 passages
Search aliases: → to
Use in source · chapters01_03.tex, line 276
Let H be the Galois group in its faithful action on four roots. Its order is divisible by four and three, so it is divisible by twelve. Since H≤ S_4, its order is twelve or twenty four. A subgroup of order twelve would have index two and would be normal. The quotient map S_4→ C_2 would send all transpositions to the same element because they are conjugate. They generate S_4, so this element must be nontrivial. Its kernel is therefore exactly A_4, the permutations expressible as a product of an e…
Exact mathematical passage
S_4\to C_2
\mapstoUse in source · 1 passage
Search aliases: ↦ mapsto
Use in source · chapters01_03.tex, line 204
[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/05_cyclic_cubic.pdf captionThe cubic realization closes inside one root field. The transformation r↦ r^2-2 cycles its three roots, and their exact symmetric identities recover the original polynomial. The group has order three because the polynomial is irreducible and already splits in that field.
Exact mathematical passage
r\mapsto r^2-2
\hookrightarrowUse in source · 2 passages
Search aliases: ↪ hookrightarrow
Use in source · chapters01_03.tex, line 52
Suppose f has d distinct roots in its splitting field L. Every automorphism of L/Q permutes them. The action is faithful because the roots generate L: an automorphism fixing every root fixes every rational expression in those roots. Consequently Gal(L/Q)↪ S_d. If f is irreducible, the action is transitive. One way to see this is to send any root to any other through the isomorphism between the corresponding simple extensions, then extend that embedding to the splitting field. Normality ensures t…
Exact mathematical passage
\label{eq:g01-root-action} \Gal(L/\Q)\hookrightarrow S_d.
\notinUse in source · 3 passages
Search aliases: ∉ notin
Use in source · chapters01_03.tex, line 222
The field Q(α) is contained in R, whereas ζ is nonreal. Thus ζ∉Q(α). Its quadratic equation remains irreducible over this real field, and [L:Q]=[L:Q(α)] [Q(α):Q]=2·3=6. The faithful action on three roots embeds Gal(L/Q) into S_3. Both groups have order six, so the embedding is an isomorphism.
Exact mathematical passage
\zeta\notin\Q(\alpha)
\prodUse in source · 4 passages
Search aliases: ∏ prod
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
V=\prod_{i<j}(r_i-r_j)
\prod_{g\in G}Use in source · 1 passage
Search aliases: ∏_g∈ G prod in
Use in source · chapters01_03.tex, line 345
In our particular construction, a=x_1 has |G| distinct images. The polynomial P(T)=∏_g∈ G(T-x_g) has degree |G| and coefficients in K. Any polynomial over K vanishing at x_1 vanishes at every x_g after applying the automorphisms. It therefore has at least |G| distinct roots. The minimal polynomial of x_1 has degree |G|. In view of~[source reference], E=K(x_1). This establishes a primitive element explicitly for the generic extension. The labels on its roots still encode the regular action.
Exact mathematical passage
\label{eq:g03-orbit-polynomial} P(T)=\prod_{g\in G}(T-x_g)
\prod_{h\in G}Use in source · 1 passage
Search aliases: ∏_h∈ G prod in
Use in source · chapters01_03.tex, line 339
An orbit polynomial gives a useful piece of the theorem's mechanism. For any a∈ E, P_a(T)=∏_h∈ G(T-σ_h(a)) has coefficients in K, since the action permutes its factors. Thus every element of E is algebraic over K. Orbit repetitions are allowed, so this polynomial need not be the minimal polynomial and can have repeated factors. The fact that all such elements together generate a finite extension of degree exactly |G| is the content of Artin's theorem. Bounded degrees for individual elements shou…
Exact mathematical passage
P_a(T)=\prod_{h\in G}(T-\sigma_h(a))
\prod_{h\in H}Use in source · 1 passage
Search aliases: ∏_h∈ H prod in
Use in source · chapters04_06.tex, line 248
In the given finite Galois extension, the normalized fixed projection is P_H(x)=frac1|H|∑_h∈ Hh(x) =frac1|H|Tr_L/L^H(x). The norm is N_L/L^H(x)=∏_h∈ Hh(x). The sum and product descriptions, together with their tower identities, are classical inputs [source reference]. They show where the resolved formulation has a linear transfer and where it needs a nonlinear transfer.
Exact mathematical passage
\operatorname{N}_{L/L^H}(x)=\prod_{h\in H}h(x). \label{eq:b-norm-product}
\prod_{i<j}Use in source · 1 passage
Search aliases: ∏_i<j prod
Use in source · chapters01_03.tex, line 208
The discriminant offers a second group check. If r_1,r_2,r_3 are the roots, the Vandermonde product V=∏_i<j(r_i-r_j) satisfies V^2=49. In a field of characteristic zero, this implies V=7 or V=-7, so every rational automorphism fixes V. Permuting the roots multiplies V by the sign of the permutation. The group is therefore contained in A_3. Irreducibility makes it transitive, forcing it to be A_3≃ C_3. This is a useful cross check of the explicit field computation. The figure labels the discrimin…
Exact mathematical passage
V=\prod_{i<j}(r_i-r_j)
\sum_gUse in source · 3 passages
Search aliases: ∑_g sum
Use in source · chapters01_03.tex, line 361
Already the invariant s=∑_g∈ Gx_g is transcendental over Q. If a nonzero polynomial F∈Q[T] satisfied F(s)=0, then F(∑_gx_g) would be a zero polynomial in independent variables. Substituting zero for every variable except x_1 would give the impossible identity F(x_1)=0. Thus s∈ K∖Q.
Exact mathematical passage
F(\sum_gx_g)
\sum_{h,k\in H}Use in source · 1 passage
Search aliases: ∑_h,k∈ H sum in
Use in source · chapters04_06.tex, line 48
Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed vector is unchanged by every summand, and therefore by the average.
Exact mathematical passage
E_H^2=\frac1{|H|^2}\sum_{h,k\in H}\lambda(hk) =\frac1{|H|}\sum_{u\in H}\lambda(u)=E_H.
\sum_{h\in H}Use in source · 4 passages
Search aliases: ∑_h∈ H sum in
Use in source · appendices.tex, line 8
The group-algebra arguments use the left regular module over C, with a rational matrix model in the finite examples. If λ(g)e_x=e_gx, the orbits of the left H action are the right cosets Hx. The normalized average is E_H=|H|^-1∑_h∈ Hλ(h). In the field-action formulation the same averaging operator is sometimes denoted P_H. A normal basis identifies the modules only after a finite Galois field has been supplied.
Exact mathematical passage
E_H=|H|^{-1}\sum_{h\in H}\lambda(h)
\sum_{u\in H}Use in source · 1 passage
Search aliases: ∑_u∈ H sum in
Use in source · chapters04_06.tex, line 48
Inversion permutes the elements of H, so E_H^*=E_H. Every element u∈ H occurs exactly |H| times among the products hk with h,k∈ H. Consequently E_H^2=frac1|H|^2∑_h,k∈ Hλ(hk) =frac1|H|∑_u∈ Hλ(u)=E_H. For a∈ H, left multiplication permutes the terms in the average, giving λ(a)E_H=E_H. Thus its range consists of fixed vectors. Conversely, an H fixed vector is unchanged by every summand, and therefore by the average.
Exact mathematical passage
E_H^2=\frac1{|H|^2}\sum_{h,k\in H}\lambda(hk) =\frac1{|H|}\sum_{u\in H}\lambda(u)=E_H.
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · chapters01_03.tex, line 361
Already the invariant s=∑_g∈ Gx_g is transcendental over Q. If a nonzero polynomial F∈Q[T] satisfied F(s)=0, then F(∑_gx_g) would be a zero polynomial in independent variables. Substituting zero for every variable except x_1 would give the impossible identity F(x_1)=0. Thus s∈ K∖Q.
Exact mathematical passage
s\in K\setminus\Q
\circUse in source · 3 passages
Search aliases: ∘ circ
Use in source · chapters04_06.tex, line 255
the field tower is L/L^H/L^K. Trace and norm compose through that tower. One way to see the formulas is to choose coset representatives for K/H, multiply or sum first over H, and then over the representatives. Every element of K appears once. Although H need not be normal in K, the embeddings of L^H into L over L^K are represented by these cosets. This gives Tr_L/L^K =Tr_L^H/L^K∘Tr_L/L^H, N_L/L^K =N_L^H/L^K∘N_L/L^H. The norm is homogeneous of degree |H| over the fixed field. It is multiplicative…
Exact mathematical passage
\operatorname{Tr}_{L/L^K} =\operatorname{Tr}_{L^H/L^K}\circ\operatorname{Tr}_{L/L^H}, \qquad \operatorname{N}_{L/L^K} =\operatorname{N}_{L^H/L^K}\circ\operatorname{N}_{L/L^H}.
\simeqUse in source · 7 passages
Search aliases: ≃ simeq
Use in source · chapters01_03.tex, line 11
[Inverse Galois over the rational numbers] For every finite group G, there is a finite Galois extension L/Q and an isomorphism G≃Gal(L/Q).
Exact mathematical passage
G\simeq\Gal(L/\Q).
\congUse in source · 2 passages
Search aliases: ≅ cong
Use in source · chapters04_06.tex, line 215
ndence is an external input, in the form stated by Milne [source reference]. It sends H≤ G to L^H={x∈ L:h(x)=x for every h∈ H}. This is a field because every h respects sums, products, and inverses. The theorem identifies Gal(L/L^H)=H, gives [L^H:Q]=[G:H], and makes subgroup inclusion reverse field inclusion. If H is normal, it also gives Gal(L^H/Q)≅ G/H.
Exact mathematical passage
\Gal(L^H/\Q)\cong G/H
\equivUse in source · 2 passages
Search aliases: ≡ equiv
Use in source · chapters07_08.tex, line 23
The group law must still hold. With our convention that σ_gσ_h means apply σ_h first, the required congruences are r_1(X) · ≡ Xpmod f, r_h(r_g(X)) · ≡ r_gh(X)pmod f. The reversed order of the displayed polynomial substitutions is deliberate. Applying σ_g to r_h(θ) replaces θ by r_g(θ), since rational coefficients are fixed. Finally the residues r_gbmod f must be pairwise distinct. Equal generator images would give equal automorphisms and could collapse the intended group.
Exact mathematical passage
r_1(X)&\equiv X\pmod f,\label{eq:g07-unit}\\ r_h(r_g(X))&\equiv r_{gh}(X)\pmod f.\label{eq:g07-composition}
\leUse in source · 2 passages
Search aliases: ≤ le
Use in source · chapters01_03.tex, line 63
[tbp] centering includegraphics[width=0.96linewidth]../visual/figures/02_field_group.pdf captionFor an already supplied finite Galois extension L/K with group G, subgroups H≤ G correspond to intermediate fixed fields L^H. The direction of inclusion reverses. The classical correspondence computes degrees and organizes fields after the ambient field extension has been established [source reference].
Exact mathematical passage
H\le G
\leqUse in source · 7 passages
Search aliases: ≤ leq
Use in source · chapters04_06.tex, line 23
For H≤ G, define e_H=frac1|H|∑_h∈ Hh∈C[G], E_H=λ(e_H)=frac1|H|∑_h∈ Hλ(h). The notation E_H in this chapter and its figures always means this operator. It is an average with normalization 1/|H|, rather than an unnormalized sum.
Exact mathematical passage
H\leq G
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · chapters04_06.tex, line 391
Fix an algebraic closure overlineQ, and write G_Q=Gal(overlineQ/Q) with its Krull topology. For a finite Galois subfield L⊂overlineQ, restriction supplies a continuous epimorphism [source reference] ρ:G_Q⟶ G=Gal(L/Q). Now specify an exact sequence of finite groups 1⟶ N⟶ Eoversetπ⟶G⟶1. The finite targets carry the discrete topology. A solution is a continuous homomorphism widetildeρ:G_Q⟶ E, π∘widetildeρ=ρ. It is a weak solution when this equation holds, and a proper solution when widetildeρ is al…
Exact mathematical passage
L\subset\overline\Q
\subseteqUse in source · 5 passages
Search aliases: ⊆ subseteq
Use in source · appendices.tex, line 10
Subgroup inclusion reverses fixed-field inclusion. In the regular module, H⊆ K gives ran(E_K)⊆ran(E_H). Commuting subgroup averages correspond to permutable products HK=KH in the regular representation. Group faithfulness alone does not justify the converse. The exact S_4 example in Chapter~4 demonstrates the difference.
Exact mathematical passage
H\subseteq K
\otimesUse in source · 1 passage
Search aliases: ⊗ otimes
Use in source · chapters07_08.tex, line 75
The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before producing the basis. It cannot be used backwards to infer that an arbitrary choice of these coefficients has all the required properties.
Exact mathematical passage
\mu:\Q[G]\otimes_\Q\Q[G]\longrightarrow\Q[G], \qquad \mu(u,v)=\Phi^{-1}(\Phi(u)\Phi(v)).
\otimes_\QUse in source · 1 passage
Search aliases: ⊗_Q otimes Q
Use in source · chapters07_08.tex, line 75
The transported multiplication is a bilinear map μ:Q[G]⊗_QQ[G]⟶Q[G], μ(u,v)=Φ^-1(Φ(u)Φ(v)). For the left regular action λ, it obeys λ(g)μ(u,v)=μ(λ(g)u,λ(g)v). It is associative and commutative, has a unit and has no zero divisors. Its coefficients depend on the field and on the chosen normal basis. The normal basis theorem assumes that field before producing the basis. It cannot be used backwards to infer that an arbitrary choice of these coefficients has all the required properties.
Exact mathematical passage
\mu:\Q[G]\otimes_\Q\Q[G]\longrightarrow\Q[G], \qquad \mu(u,v)=\Phi^{-1}(\Phi(u)\Phi(v)).
\langleUse in source · 12 passages
Search aliases: ⟨ langle
Use in source · appendices.tex, line 43
For the ordered basis (), (23), (12), (123), (132), (13) the script constructs matrix columns from λ(g)e_x=e_gx. Every entry is an exact rational number. The H=⟨(12)⟩ and K=⟨(23)⟩ commutator is nonzero. Matrix ranks are computed over Q, not with a floating tolerance. The figure source reads the frozen JSON generated by the same run. The individual roots plotted in the complex plane are numerical illustrations, not group certificates.
Exact mathematical passage
H=\langle(12)\rangle
\rangleUse in source · 12 passages
Search aliases: ⟩ rangle
Use in source · appendices.tex, line 43
For the ordered basis (), (23), (12), (123), (132), (13) the script constructs matrix columns from λ(g)e_x=e_gx. Every entry is an exact rational number. The H=⟨(12)⟩ and K=⟨(23)⟩ commutator is nonzero. Matrix ranks are computed over Q, not with a floating tolerance. The figure source reads the frozen JSON generated by the same run. The individual roots plotted in the complex plane are numerical illustrations, not group certificates.
Exact mathematical passage
H=\langle(12)\rangle
\longrightarrowUse in source · 6 passages
Search aliases: ⟶ longrightarrow
Use in source · chapters04_06.tex, line 230
The normal basis theorem supplies a useful translation [source reference]. For some α∈ L, the elements g(α), with g∈ G, form a Q basis. The map Q[G]⟶ L, ∑_g c_g g⟼∑_g c_g g(α) is an equivariant vector space isomorphism. After extending scalars to C, it transports the regular subgroup averages to the fixed vector spaces of the field action. This verifies the Galois to resolution direction with all its subgroup information preserved.
Exact mathematical passage
\Q[G]\longrightarrow L, \qquad \sum_g c_g g\longmapsto\sum_g c_g g(\alpha) \label{eq:b-normal-basis-map}
\longmapstoUse in source · 2 passages
Search aliases: ⟼ longmapsto
Use in source · chapters04_06.tex, line 230
The normal basis theorem supplies a useful translation [source reference]. For some α∈ L, the elements g(α), with g∈ G, form a Q basis. The map Q[G]⟶ L, ∑_g c_g g⟼∑_g c_g g(α) is an equivariant vector space isomorphism. After extending scalars to C, it transports the regular subgroup averages to the fixed vector spaces of the field action. This verifies the Galois to resolution direction with all its subgroup information preserved.
Exact mathematical passage
\Q[G]\longrightarrow L, \qquad \sum_g c_g g\longmapsto\sum_g c_g g(\alpha) \label{eq:b-normal-basis-map}
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