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Direct sum of two Artin representations
Direct difference of two Artin representations
Tensor product of two Artin representations
True iff the two Artin representations are equal
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);
> triv,sign,rho:=Explode(ArtinRepresentations(K));
> 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
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 $.1^4 - 10*$.1^2 + 1
over the Rational Field
> _,_,sign3:=MinimalField(twist);
> sign3;
Artin representation of Number Field with defining polynomial
$.1^2 - 10*$.1 + 1 over the Rational Field with character
( 1, -1 ) and conductor 24
> sign1*sign2*sign3 eq triv1;
true
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