FiniteFieldElementTrace

FiniteFieldElementTrace[a]

gives the absolute trace of the finite field element a.

FiniteFieldElementTrace[a,k]

gives the trace of a relative to the -element subfield of the ambient field of a.

FiniteFieldElementTrace[a,emb]

gives the trace of a relative to the finite field embedding emb.

Details

  • For a finite field with characteristic p and extension degree d over , the absolute trace of a is given by . is a -linear mapping from to .
  • If MinimalPolynomial[a,x]xn+cn-1xn-1+⋯+c0, then .
  • FiniteFieldElementTrace[a] gives an integer between and .
  • For a finite field with characteristic p and extension degree d over , the trace of a relative to the -element subfield of is given by , where . is a -linear mapping from to . k needs to be a divisor of d.
  • If MinimalPolynomial[a,x,k]xn+cn-1xn-1+⋯+c0, then .
  • FiniteFieldElementTrace[a,k] gives an element of .
  • If emb=FiniteFieldEmbedding[e1e2], then FiniteFieldElementTrace[a,emb] effectively gives emb["Projection"][FiniteFieldElementTrace[a,k]], where a belongs to the ambient field of e2 and k is the extension degree of the ambient field of e1.

Examples

open allclose all

Basic Examples  (1)

Represent a finite field with characteristic and extension degree :

Find the absolute trace of an element of the field:

Find the trace relative to the -element subfield:

Scope  (2)

Find the absolute trace of a finite field element:

The absolute trace given as a finite field element:

The trace relative to the -element subfield:

Compute the trace relative to a field embedding:

The result is equivalent to computing the trace relative to and projecting it to :

Applications  (1)

Define -linear mappings . Every -linear mapping from to has this form:

Illustrate linearity of :

Properties & Relations  (7)

is a -linear mapping from to :

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

Use FrobeniusAutomorphism to compute the conjugates of a:

The absolute trace of is equal to the absolute trace of :

If is the -element subfield of , then is a -linear mapping from to :

Use FiniteFieldEmbedding to embed an -element field in :

Since , this shows that c and d belong to :

This illustrates -linearity of :

Construct field embeddings such that :

FiniteFieldElementTrace satisfies a transitivity property:

If MinimalPolynomial[a,x]xn+cn-1xn-1+⋯+c0, then :

If MinimalPolynomial[a,x,k]xn+cn-1xn-1+⋯+c0, then :

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

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2024_finitefieldelementtrace, author="Wolfram Research", title="{FiniteFieldElementTrace}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteFieldElementTrace.html}", note=[Accessed: 01-May-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_finitefieldelementtrace, organization={Wolfram Research}, title={FiniteFieldElementTrace}, year={2023}, url={https://reference.wolfram.com/language/ref/FiniteFieldElementTrace.html}, note=[Accessed: 01-May-2024 ]}