Arithmetic

A1 + A2: ArtRep, ArtRep -> ArtRep: Direct sum of two Artin representations

A1 - A2: ArtRep, ArtRep -> ArtRep: Direct difference of two Artin representations

A1 * A2: ArtRep, ArtRep -> ArtRep: Tensor product of two Artin representations

A1 eq A2: ArtRep, ArtRep -> BoolElt: Returns true iff the two Artin representations are equal

A1 ne A2: ArtRep, ArtRep -> BoolElt

Returns true iff the two Artin representations are not equal

Sums, differences, and tensor products of relative Artin representations attached to the same number field are relative Artin representations over the same base field. Arithmetic between representations attached to different number fields, which passes to the compositum, is available only over \(\Q\).

Example: Arithmetic of Artin Representations (ex-ba9cf8)

For Artin representations constructed from the same number field, their arithmetic is just arithmetic of characters:

> P<x> := PolynomialRing(Rationals());
> K := NumberField(x^3-2);
> A := ArtinRepresentations(K: Ramification:=true);
> triv, sign, rho := Explode(A);
> triv;
Artin representation S3: (1,1,1) of ext<Q|x^3-2>, conductor 1
> rho;
Artin representation S3: (2,0,-1) of ext<Q|x^3-2>, conductor 108
> triv+rho;
Artin representation S3: (3,1,0) of ext<Q|x^3-2>, conductor 108
> sign*rho eq rho;
true

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Example: Artin Representations over Different Fields (ex-401593)

When Artin representations factor through different fields, their arithmetic involves the compositum of the fields:

> K1 := QuadraticField(2);
> triv1, sign1 := Explode(ArtinRepresentations(K1));
> K2 := QuadraticField(3);
> triv2, sign2 := Explode(ArtinRepresentations(K2));
> twist := sign1*sign2;
> Field(twist);
Number Field with defining polynomial x^4 - 10*x^2 + 1 over the Rational Field
> sign3 := Minimize(twist);
> sign3;
Artin representation C2: (1,-1) of ext<Q|x^2-6>
> sign1*sign2*sign3 eq triv1;
true

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Arithmetic#

Arithmetic