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Compute all irreducible Artin representations that factor through the normal
closure of K.
Writing F for the normal closure of K/Q, this function converts
an abstract group character of Gal(F/Q) into an Artin representation.
F: FldNum Default: normal closure of K
Embedding: Map Default:
Construct the permutation representation A of the absolute Galois group of
Q on the embeddings of K into C. This is an Artin representation
of Gal(F/Q) of dimension [K:Q], where
by default F := Field(A) is the normal closure of K.
It is possible to specify F to be any Galois extension
of Q containing K. Unless there is a default coercion,
an explicit embedding of K into F must be given
with the Embedding parameter.
Embedding: Map Default:
Given an Artin representation A and an (optional) embedding
of Field(A) into F, lift A to F.
A quadratic field K has two irreducible Artin representations the factor
through Gal(K/Q), the trivial one and the quadratic character of K:
> K<i>:=QuadraticField(-1);
> triv,sign:=Explode(ArtinRepresentations(K));
> sign;
Artin representation of Quadratic Field with defining polynomial x^2 + 1
over the Rational Field with character ( 1, -1 ) and conductor 4
An alternative way to define them is directly by their character:
> triv,sign:Magma;
QuadraticField(-1) !! [1,1]
QuadraticField(-1) !! [1,-1]
The regular representation of Gal(K/Q) is their sum:
> PermutationCharacter(K);
Artin representation of Quadratic Field with defining
polynomial z^2 + 1 over the Rational Field with
character ( 2, 0 ) and conductor 4
> $1 eq triv+sign;
true
Next, let L=K(Sqrt( - 2 - i)). Then L has normal
closure F with Gal(F/Q)=D4, the dihedral group of order 8:
> L:=ext<K|Polynomial([2+i,0,1])>;
> G:=GaloisGroup(AbsoluteField(L));
> IsIsomorphic(G,DihedralGroup(4));
true
> [Dimension(A): A in ArtinRepresentations(L)];
[1, 1, 1, 1, 2 ]
We use BaseChange to lift Artin representations from Gal(K/Q) to
Gal(F/Q), and Kernel to descend back to a smaller field.
> A:=BaseChange(sign,L);
> A;
Artin representation of Number Field with defining
polynomial $.1^2 + i + 2 over its ground field with
character ( 1, 1, -1, 1, -1 ) and conductor 4
> _,_,sign2:=Kernel(A);
> sign2 eq sign;
true;
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