We take an extension F of K=Q
3 with Galois group F
5=C
5:C
4 of order 20.
It has five irreducible representations. Four of them are 1-dimensional, and
so they actually factor through a smaller Galois group (C
1, C
2 or C
4).
Minimize descends them to these Galois groups, although they are of course
still the same as representations of the absolute Galois group.
> K:=pAdicField(3,20);
> R<x>:=PolynomialRing(K);
> list:=GaloisRepresentations(x^5-3);
> list;
[
1-dim trivial Galois representation 1 over Q3[20],
1-dim unramified Galois representation (1,1,-1,-1,1) with G=F5, I=C5
over Q3[20],
1-dim unramified Galois representation (1,-1,-zeta(4)_4,zeta(4)_4,1)
with G=F5, I=C5 over Q3[20],
1-dim unramified Galois representation (1,-1,zeta(4)_4,-zeta(4)_4,1)
with G=F5, I=C5 over Q3[20],
4-dim Galois representation (4,0,0,0,-1) with G=F5, I=C5, conductor 3^4
over Q3[20]
]
> [Minimize(A): A in list];
[
1-dim trivial Galois representation 1 over Q3[20],
1-dim unramified Galois representation (1,-1) with G=C2, I=C1 over Q3[20],
1-dim unramified Galois representation (1,-1,-zeta(4)_4,zeta(4)_4)
with G=C4, I=C1 over Q3[20],
1-dim unramified Galois representation (1,-1,zeta(4)_4,-zeta(4)_4)
with G=C4, I=C1 over Q3[20],
4-dim Galois representation (4,0,0,0,-1) with G=F5, I=C5, conductor 3^4
over Q3[20]
]
> forall{A: A in list | A eq Minimize(A)};
true
Finally, we illustrate how the parameter To may be used to descend a Galois
representation to a specific Galois group, in this case the Galois group
Gal(F/K) isomorphic to C
4 of the degree 4 unramified extension of K.
> F:=ext<K|4>; // Take F = degree 4 unr. ext. of K, and
> B:=PermutationCharacter(F,K); // any B with BaseField(B)=K, Field(B)=F
> list[2];
1-dim unramified Galois representation (1,1,-1,-1,1) with G=F5, I=C5 over Q3[20]
> Minimize(list[2]: To:=B);
1-dim unramified Galois representation (1,1,-1,-1) with G=C4, I=C1 over Q3[20]
> Minimize(list[2]);
1-dim unramified Galois representation (1,-1) with G=C2, I=C1 over Q3[20]
BaseChange(A,L) : GalRep,FldPad -> GalRep
Base change (restriction) of a Galois representation A over K over a finite
extension L/K.
We take a 2-dimensional irreducible representation of
Gal(Q
2(ζ
3, root 3 of 2)) isomorphic to S
3 and check that
its base change to Q
2(ζ
3) is reducible.
> K:=pAdicField(2,20);
> R<x>:=PolynomialRing(K);
> A:=GaloisRepresentations(x^3-2)[3]; A;
2-dim Galois representation (2,0,-1) with G=S3, I=C3, conductor 2^2 over Q2[20]
> L:=ext<K|2>;
> R:=Restriction(A,L); R;
2-dim Galois representation (2,-1,-1) with G=C3, I=C3, conductor 2^2
over ext<Q2[20]|2>
> IsIrreducible(A),IsIrreducible(R);
true false
Induction of a Galois representation A over K to a subfield K0⊂K.
> K0:=pAdicField(2,20); // K0=Q2
> K:=ext<K0|2>; // K=Q2(zeta_3)
> R<x>:=PolynomialRing(K);
> A:=GaloisRepresentations(x^3-102)[3];
> A; // 1-dim character over K
1-dim Galois representation (1,-zeta(3)_3-1,zeta(3)_3) with G=C3, I=C3,
conductor 2^1 over ext<Q2[20]|2>
> Induction(A,K0); // Induced to K0
2-dim Galois representation (2,0,-1) with G=S3, I=C3, conductor 2^2 over Q2[20]
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