gives the norm of a relative to the 
-element subfield of the ambient field of a.
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 all close allBasic Examples (1)
Represent a finite field with characteristic 
and extension degree 
:
ℱ = FiniteField[17, 6]Find the absolute norm of an element of the field:
FiniteFieldElementNorm[ℱ[123]]Find the norm relative to the 
-element subfield:
FiniteFieldElementNorm[ℱ[123], 2]Scope (2)
Find the absolute norm of a finite field element:
ℱ = FiniteField[127, 12];
FiniteFieldElementNorm[ℱ[1234]]The absolute norm given as a finite field element:
FiniteFieldElementNorm[ℱ[1234], 1]The norm relative to the 
-element subfield:
FiniteFieldElementNorm[ℱ[1234], 3]Compute the norm relative to a field embedding:
{𝒦, ℱ} = {FiniteField[73, 2], FiniteField[73, 8]};
ℰ = FiniteFieldEmbedding[𝒦, ℱ];
FiniteFieldElementNorm[ℱ[1234], ℰ]The result is equivalent to computing the norm relative to 
and projecting it to 
:
FiniteFieldElementNorm[ℱ[1234], 2]ℰ["Projection"][%]Applications (1)
ℱ = FiniteField[59, 3];
b = ℱ[123];
m[b_][a_] := a b
FiniteFieldElementNorm[b]Compute the determinant manually:
Mod[Det[{PadRight[ m[b][ℱ[{1, 0, 0}]]["Coefficients"], 3], PadRight[m[b][ℱ[{0, 1, 0}]]["Coefficients"], 3], PadRight[m[b][ℱ[{0, 0, 1}]]["Coefficients"], 3]}], 59]Properties & Relations (7)
is a mapping from 
to 
which preserves multiplication:
ℱ = FiniteField[79, 3];
{a, b} = {ℱ[123], ℱ[456]};
FiniteFieldElementNorm[a b]Mod[FiniteFieldElementNorm[a] FiniteFieldElementNorm[b], 79]The absolute norm of a is equal to the product of all conjugates of a:
ℱ = FiniteField[17, 4];
a = ℱ[123];
FiniteFieldElementNorm[a]Use FrobeniusAutomorphism to compute the conjugates of a:
Product[FrobeniusAutomorphism[a, k], {k, 0, 3}]The absolute norm of 
is equal to the absolute norm of 
:
ℱ = FiniteField[101, 3];
a = ℱ[123];
FiniteFieldElementNorm[a]FiniteFieldElementNorm[a ^ 101]If 
is the 
-element subfield of 
, then 
is a mapping from 
to 
, which preserves multiplication:
ℱ = FiniteField[11, 6];
{a, b} = {ℱ[123], ℱ[456]};
{c, d} = {FiniteFieldElementNorm[a, 3], FiniteFieldElementNorm[b, 3]}Use MinimalPolynomial to show that c and d belong to the 
-element subfield 
of 
:
Exponent[{MinimalPolynomial[c, x], MinimalPolynomial[d, x]}, x]This illustrates the multiplication-preserving property of 
:
FiniteFieldElementNorm[a b, 3]c dConstruct field embeddings such that 
:
𝒦 = FiniteField[71, 3];
ℱ = FiniteField[71, 6];
𝒢 = FiniteField[71, 12];
Subscript[ℰ, 1] = FiniteFieldEmbedding[𝒦, ℱ];
Subscript[ℰ, 2] = FiniteFieldEmbedding[ℱ, 𝒢];
Subscript[ℰ, 21] = Subscript[ℰ, 2] @* Subscript[ℰ, 1];
Subscript[ℰ, 21][𝒦[123]] == Subscript[ℰ, 2][Subscript[ℰ, 1][𝒦[123]]]FiniteFieldElementNorm satisfies a transitivity property:
FiniteFieldElementNorm[𝒢[1234], Subscript[ℰ, 21]]FiniteFieldElementNorm[FiniteFieldElementNorm[𝒢[1234], Subscript[ℰ, 2]], Subscript[ℰ, 1]]If MinimalPolynomial[a,x]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[149, 5];
a = ℱ[1234];
FiniteFieldElementNorm[a]MinimalPolynomial[a, x]Mod[-Coefficient[%, x, 0], 149]If MinimalPolynomial[a,x,k]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[43, 6];
a = ℱ[1234];
FiniteFieldElementNorm[a, 2]MinimalPolynomial[a, x, 2]-Coefficient[%, x, 0]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