Inverse Galois: source and evidence

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

Read the typeset paper

1 \chapter{Conventions, source contracts and exact runs}
2 \label{app:g09-evidence}
3 
4 \section{Notation and coefficient domains}
5 
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 $E=\Q(x_g:g\in G)$ in Chapter~\ref{chap:g03-generic} 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.
7 
8 The group-algebra arguments use the left regular module over $\C$, with a rational matrix model in the finite examples. If $\lambda(g)e_x=e_{gx}$, the orbits of the left $H$ action are the right cosets $Hx$. The normalized average is $E_H=|H|^{-1}\sum_{h\in H}\lambda(h)$. 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.
9 
10 Subgroup inclusion reverses fixed-field inclusion. In the regular module, $H\subseteq K$ gives $\operatorname{ran}(E_K)\subseteq\operatorname{ran}(E_H)$. 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.
11 
12 An arithmetic embedding problem uses the profinite group $G_{\Q}=\Gal(\overline{\Q}/\Q)$ 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.
13 
14 \section{External inputs retained by the proof chain}
15 
16 \begin{longtable}{@{}p{.12\linewidth}p{.79\linewidth}@{}}
17 \toprule Contract & Exact input and scope\\\midrule\endhead
18 LC03 & Hilbert specialization over $\Q$, with the full primitive extension and nonzero exceptional-locus restrictions. One parameter and several parameters have different excluded sets.\\[4pt]
19 LC04 & Every finite solvable group occurs over $\Q$, from the corrected Shafarevich proof source. Additional local prescriptions are outside this statement.\\[4pt]
20 LC08 & Good-prime squarefree factor degrees yield a cycle in the characteristic-zero splitting-field group of a monic integral polynomial.\\[4pt]
21 LC09 & Irreducible cubic discriminant classification over $\Q$. The characteristic restrictions in the source are recorded explicitly.\\[4pt]
22 LC10 & Artin's finite automorphism theorem for an actual field with faithful finite automorphism group.\\[4pt]
23 LC11 & The subgroup/fixed-field correspondence for an extension already known to be finite Galois.\\[4pt]
24 LC12 & Normal basis for a finite Galois extension. This is a vector-space identification, not a field construction.\\[4pt]
25 LC13 & Trace and norm formulas and tower composition for the separable extensions used here.\\[4pt]
26 LC14 & Versioned $M_{23}$ preprint context, outside the elementary proof dependencies and not independently computationally reproduced.\\[4pt]
27 LC15 & Noether rationality as a sufficient conditional route. Contextual negative examples are source-attributed rather than independently audited here.\\[4pt]
28 LC16 & Continuous finite quotients of the absolute Galois group correspond to finite Galois field extensions.\\[4pt]
29 LC17 & Every existing finite separable extension has a primitive element, so a completed characteristic-zero realization admits the certificate format.\\
30 \bottomrule
31 \end{longtable}
32 
33 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.
34 
35 \section{The finite computation actually executed}
36 
37 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.
38 
39 For the ordered basis
40 \[
41  (),\ (23),\ (12),\ (123),\ (132),\ (13)
42 \]
43 the script constructs matrix columns from $\lambda(g)e_x=e_{gx}$. Every entry is an exact rational number. The $H=\langle(12)\rangle$ and $K=\langle(23)\rangle$ 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.
44 
45 The bounded execution wrapper records the actual command, software versions, input hashes before and after execution, runtime, resource limits and output hashes in a version~2 manifest. Its validator passed with the project root supplied. An independent reviewer reconstructed the subgroup matrices from coset incidence and enumerated the subsets of $S_3$ to check that the stated six-subgroup range is complete. These checks support the finite records. The universal projector theorem still uses its written argument.
46 
47 \section{Reproduction}
48 
49 From the extracted bundle root, the portable exact calculation is
50 \begin{lstlisting}
51 python3 computations/exact_examples.py \
52   --output computations/exact_examples.json
53 \end{lstlisting}
54 Python and SymPy are required. The bounded provenance wrapper additionally requires a POSIX environment for its enforced limits:
55 \begin{lstlisting}
56 python3 computations/run_examples.py
57 \end{lstlisting}
58 That wrapper creates the evidence logs, manifest and scientific JSON, then copies the same JSON bytes to the figure input path. It does not depend on a random seed. The shipped manifest records the versions actually used, rather than guaranteeing byte-identical serialized output for all future library versions.
59 
60 Rebuild the readers with
61 \begin{lstlisting}
62 python3 build.py --figures
63 \end{lstlisting}
64 The PDF and scientific plotting dependencies are listed in the bundle README. Rebuilding can change PDF hashes or page flow. The delivered page-review records apply to the delivered PDFs, and a changed build needs renewed visual inspection. Compilation alone is not a layout review or a mathematical proof.
65 
66 The four original recovered inputs are retained in \texttt{historical/}. Their byte counts and hashes are checked separately. Historical audits retain their original scope. The current manuscript, exact calculations, source contracts, proof review and page reviews supply this edition's claim boundary.

Back to source controls

Back to Inverse Galois: source and evidence