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

Example: Artin Arith1 (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 Arith2 (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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