gives the trace of a relative to the 
-element subfield of the ambient field of a.
FiniteFieldElementTrace
gives the absolute trace of the finite field element a.
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 all close allBasic Examples (1)
Represent a finite field with characteristic 
and extension degree 
:
ℱ = FiniteField[17, 6]Find the absolute trace of an element of the field:
FiniteFieldElementTrace[ℱ[123]]Find the trace relative to the 
-element subfield:
FiniteFieldElementTrace[ℱ[123], 2]Scope (2)
Find the absolute trace of a finite field element:
ℱ = FiniteField[127, 12];
FiniteFieldElementTrace[ℱ[1234]]The absolute trace given as a finite field element:
FiniteFieldElementTrace[ℱ[1234], 1]The trace relative to the 
-element subfield:
FiniteFieldElementTrace[ℱ[1234], 3]Compute the trace relative to a field embedding:
{𝒦, ℱ} = {FiniteField[73, 2], FiniteField[73, 8]};
ℰ = FiniteFieldEmbedding[𝒦, ℱ];
FiniteFieldElementTrace[ℱ[1234], ℰ]The result is equivalent to computing the trace relative to 
and projecting it to 
:
FiniteFieldElementTrace[ℱ[1234], 2]ℰ["Projection"][%]Applications (1)
Properties & Relations (7)
is a 
-linear mapping from 
to 
:
ℱ = FiniteField[29, 3];
{a, b} = {ℱ[123], ℱ[456]};
FiniteFieldElementTrace[2a + 3b]Mod[2FiniteFieldElementTrace[a] + 3FiniteFieldElementTrace[b], 29]The absolute trace of a is equal to the sum of all conjugates of a:
ℱ = FiniteField[17, 4];
a = ℱ[123];
FiniteFieldElementTrace[a]Use FrobeniusAutomorphism to compute the conjugates of a:
Sum[FrobeniusAutomorphism[a, k], {k, 0, 3}]The absolute trace of 
is equal to the absolute trace of 
:
ℱ = FiniteField[101, 3];
a = ℱ[123];
FiniteFieldElementTrace[a]FiniteFieldElementTrace[a ^ 101]If 
is the 
-element subfield of 
, then 
is a 
-linear mapping from 
to 
:
ℱ = FiniteField[11, 6];
{a, b} = {ℱ[123], ℱ[456]};
{c, d} = {FiniteFieldElementTrace[a, 3], FiniteFieldElementTrace[b, 3]}Use FiniteFieldEmbedding to embed an 
-element field 
in 
:
𝒢 = FiniteField[11, 3];
ℰ = FiniteFieldEmbedding[𝒢, ℱ]Since 
, this shows that c and d belong to 
:
{ℰ[ℰ["Projection"][c]] == c, ℰ[ℰ["Projection"][d]] == d}This illustrates 
-linearity of 
:
FiniteFieldElementTrace[ℰ[𝒢[321]]a + ℰ[𝒢[654]]b, 3]ℰ[𝒢[321]]c + ℰ[𝒢[654]]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]]]FiniteFieldElementTrace satisfies a transitivity property:
FiniteFieldElementTrace[𝒢[1234], Subscript[ℰ, 21]]FiniteFieldElementTrace[FiniteFieldElementTrace[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];
FiniteFieldElementTrace[a]MinimalPolynomial[a, x]Mod[-Coefficient[%, x, 4], 149]If MinimalPolynomial[a,x,k]xn+cn-1xn-1+⋯+c0, then 
:
ℱ = FiniteField[43, 6];
a = ℱ[1234];
FiniteFieldElementTrace[a, 2]MinimalPolynomial[a, x, 2]-Coefficient[%, x, 2]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