gives the absolute norm of the finite field element a.
FiniteFieldElementNorm [a,k]
gives the norm of a relative to the -element subfield of the ambient field of a.
FiniteFieldElementNorm [a,emb]
gives the norm of a relative to the finite field embedding emb.
FiniteFieldElementNorm
gives the absolute norm of the finite field element a.
FiniteFieldElementNorm [a,k]
gives the norm of a relative to the -element subfield of the ambient field of a.
FiniteFieldElementNorm [a,emb]
gives the norm of a relative to the finite field embedding emb.
Details
- For a finite field with characteristic p and extension degree d over , the absolute norm of a is given by . is a mapping from to and .
- If MinimalPolynomial [a,x]xn+cn-1xn-1+⋯+c0, then .
- FiniteFieldElementNorm [a] gives an integer between and .
- For a finite field with characteristic p and extension degree d over , the norm of a relative to the -element subfield of is given by , where . is a mapping from to and . k needs to be a divisor of d.
- If MinimalPolynomial [a,x,k]xn+cn-1xn-1+⋯+c0, then .
- FiniteFieldElementNorm [a,k] gives an element of .
- If emb=FiniteFieldEmbedding [e1e2], then FiniteFieldElementNorm [a,emb] effectively gives emb["Projection"][FiniteFieldElementNorm[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 all close allBasic Examples (1)
Represent a finite field with characteristic and extension degree :
Find the absolute norm of an element of the field:
Find the norm relative to the -element subfield:
Scope (2)
Find the absolute norm of a finite field element:
The absolute norm given as a finite field element:
The norm relative to the -element subfield:
Compute the norm relative to a field embedding:
The result is equivalent to computing the norm relative to and projecting it to :
Applications (1)
Define -linear mappings :
computes the determinant of :
Compute the determinant manually:
Properties & Relations (7)
is a mapping from to which preserves multiplication:
The absolute norm of a is equal to the product of all conjugates of a:
Use FrobeniusAutomorphism to compute the conjugates of a:
The absolute norm of is equal to the absolute norm of :
If is the -element subfield of , then is a mapping from to , which preserves multiplication:
Use MinimalPolynomial to show that c and d belong to the -element subfield of :
This illustrates the multiplication-preserving property of :
Construct field embeddings such that :
FiniteFieldElementNorm 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 :
Related Guides
History
Text
Wolfram Research (2023), FiniteFieldElementNorm, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteFieldElementNorm.html.
CMS
Wolfram Language. 2023. "FiniteFieldElementNorm." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FiniteFieldElementNorm.html.
APA
Wolfram Language. (2023). FiniteFieldElementNorm. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FiniteFieldElementNorm.html
BibTeX
@misc{reference.wolfram_2025_finitefieldelementnorm, author="Wolfram Research", title="{FiniteFieldElementNorm}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteFieldElementNorm.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_finitefieldelementnorm, organization={Wolfram Research}, title={FiniteFieldElementNorm}, year={2023}, url={https://reference.wolfram.com/language/ref/FiniteFieldElementNorm.html}, note=[Accessed: 05-December-2025]}