# FrobeniusAutomorphism

gives the value of the Frobenius automorphism at the finite field element a.

gives the value of the kth functional power of the Frobenius automorphism at a.

# Details • For a finite field with characteristic , the Frobenius automorphism is given by .
• All finite field automorphisms are functional powers of the Frobenius automorphism.
• The number of different field automorphisms of is equal to the extension degree of over .
• Any field automorphism satisfies equations and .
• If n is the degree of the MinimalPolynomial f of an element a of , then Table[FrobeniusAutomorphism[a,k],{k,n}] gives all the roots of f in .

# Examples

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## Basic Examples(1)

Represent a finite field with characteristic and extension degree :

Compute the value of the Frobenius automorphism at an element of the field:

The third functional power of the Frobenius automorphism:

## Scope(1)

Compute the value of the Frobenius automorphism at an element of a finite field:

Compute all conjugates of a:

Conjugates of a are roots of the minimal polynomial of a:

## Applications(1)

Compute the minimal polynomial of an element of a finite field:

The minimal polynomial of is the product of over all conjugates of :

Convert to integer coefficients:

Compare with the result obtained using the built-in MinimalPolynomial:

## Properties & Relations(5)

Frobenius automorphism is a field automorphism:

For a finite field with characteristic , the Frobenius automorphism is given by :

All finite field automorphisms are functional powers of the Frobenius automorphism:

Use FiniteFieldEmbedding to find an automorphism of :

Identify the functional power of the Frobenius automorphism that gives the same mapping:

The number of different field automorphisms of is equal to the extension degree of over :

Compute all conjugates of a finite field element a:

The absolute trace of a is equal to the sum of conjugates:

Use FiniteFieldElementTrace to compute the absolute trace:

The absolute norm of a is equal to the product of conjugates:

Use FiniteFieldElementNorm to compute the absolute norm:

The conjugates are roots of :