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:

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:

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:

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 MinimalPolynomial[a]: