FrobeniusAutomorphism

FrobeniusAutomorphism[a]

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

FrobeniusAutomorphism[a,k]

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 MinimalPolynomial[a]:

Wolfram Research (2023), FrobeniusAutomorphism, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

Text

Wolfram Research (2023), FrobeniusAutomorphism, Wolfram Language function, https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

CMS

Wolfram Language. 2023. "FrobeniusAutomorphism." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html.

APA

Wolfram Language. (2023). FrobeniusAutomorphism. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html

BibTeX

@misc{reference.wolfram_2024_frobeniusautomorphism, author="Wolfram Research", title="{FrobeniusAutomorphism}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html}", note=[Accessed: 21-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_frobeniusautomorphism, organization={Wolfram Research}, title={FrobeniusAutomorphism}, year={2023}, url={https://reference.wolfram.com/language/ref/FrobeniusAutomorphism.html}, note=[Accessed: 21-April-2024 ]}