Direct sum of two Artin representations
Direct difference of two Artin representations
Tensor product of two Artin representations
Returns true iff the two Artin representations are equal
Returns true iff the two Artin representations are not equal
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
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
V2.28, 13 July 2023