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
> P<x> := PolynomialRing(Rationals()); > K := NumberField(x^3-2); > A := ArtinRepresentations(K: Ramification:=true); > triv, sign, rho := Explode(A); > triv; Artin representation of Number Field with defining polynomial x^3 - 2 over the Rational Field with character ( 1, 1, 1 ) and conductor 1 > rho; Artin representation of Number Field with defining polynomial x^3 - 2 over the Rational Field with character ( 2, 0, -1 ) and conductor 108 > triv+rho; Artin representation of Number Field with defining polynomial x^3 - 2 over the Rational Field with character ( 3, 1, 0 ) and conductor 108 > sign*rho eq rho; true
> 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 $.1^4 - 10*$.1^2 + 1 over the Rational Field > sign3 := Minimize(twist); > sign3; Artin representation of Number Field with defining polynomial $.1^2 - 6 over the Rational Field with character ( 1, -1 ) > sign1*sign2*sign3 eq triv1; true