[Next][Prev] [Right] [Left] [Up] [Index] [Root]

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 ArtRep_artin-arith1 (H44E5)

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 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

Example ArtRep_artin-arith2 (H44E6)

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 := 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

 [Next][Prev] [Right] [Left] [Up] [Index] [Root]

Version: V2.19 of Wed Apr 24 15:09:57 EST 2013