FiniteFieldElementNorm
gives the absolute norm of the finite field element a.
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 allclose allBasic Examples (1)
Scope (2)
Applications (1)
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 
:
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