19. Compute Frobenius in a quadratic finite field
All examples · Glossary: finite-field Frobenius · Papers, proofs, Lean, and code
The square residues modulo five are \(0,1,4\). Since \(3\) is absent, \(t^2-3\) is irreducible over \(\mathbb F_5\). We can form the field
\[\mathbb F_{25}=\mathbb F_5[\alpha],\qquad \alpha^2=3.
\]
Every element has a unique form \(a+b\alpha\) with \(a,b\) modulo five. This provides finite-field background for Paper XI’s finite packet theory.
Multiply in the declared basis
Use the basis \((1,\alpha)\). The rule is
\[(a+b\alpha)(c+d\alpha)=(ac+3bd)+(ad+bc)\alpha,
\]
with both coefficients reduced modulo five. For \(\beta=1+\alpha\), \(\beta^2=4+2\alpha\).
Calculate Frobenius and its orbit
The characteristic-five binomial identity gives \((a+b\alpha)^5=a^5+b^5\alpha^5\). In the base field \(a^5=a\), and
\[\alpha^5=\alpha(\alpha^2)^2=9\alpha=4\alpha=-\alpha.
\]
Consequently \(F(a+b\alpha)=a-b\alpha\). Applying \(F\) twice is the identity. The element \(\beta\) has the two-element orbit \(1+\alpha,1-\alpha\).
Compute trace, norm and inverse
\[\mathrm{Tr}(\beta)=\beta+\beta^5=2,\qquad
\mathrm{N}(\beta)=\beta\beta^5=1-3=3\quad\text{in }\mathbb F_5.
\]
Since the inverse of \(3\) modulo five is \(2\),
\[\beta^{-1}=2(1-\alpha)=2+3\alpha.
\]
Indeed, \((1+\alpha)(2+3\alpha)=11+5\alpha=1\) in this field.
Verify the operator comparison
Multiplication by \(\beta\) and Frobenius have matrices
\[M_\beta=\begin{pmatrix}1&3\\1&1\end{pmatrix},\qquad
C=\begin{pmatrix}1&0\\0&4\end{pmatrix},\qquad C^{-1}=C.
\]
Direct multiplication modulo five gives
\[CM_\beta C^{-1}=\begin{pmatrix}1&2\\4&1\end{pmatrix}=M_{\beta^5}.
\]
The matrix trace is \(2\) and its determinant is \(1-3=3\), agreeing with the field trace and norm. This constructs and checks a particular finite comparison map. Paper XI’s arithmetic realization and comparison obligations require their own supplied objects and hypotheses.
Try it
For \(\gamma=2+\alpha\), find its Frobenius conjugate, trace, norm and inverse.
Answer: \(\gamma^5=2-\alpha\), trace \(4\), norm \(4-3=1\), and \(\gamma^{-1}=2-\alpha\). All scalars in this answer are read modulo five.
Papers, proofs, Lean, and code
Terms in this example: Prime field · Extension field · Finite-field Frobenius · Field trace · Field norm · Irreducible polynomial
This is finite-field background for Paper XI’s finite packet theory. Conjecture 66 separately asks for an actual arithmetic realizer with its declared comparison data.
Paper XI: Packets, Arithmetic Realization, and Conditional Comparison
Read · Written proofs · Lean coverage · Code and setup · Evidence
Source passages:
Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: finite_frobenius(). This checks the displayed rational, modular or finite-set calculations exactly.
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