# 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 .
• 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 .
• 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

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## Basic 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 :