Galois Module Structure

If the base field k for class field constructions is normal with respect to some subfield k_0, i.e. k/k_0 is normal with Galois group G and if the defining modulus of the ideal group is G--invariant, then G acts on the ideal group. The following functions view ideal groups as Galois modules. Given an abelian extension A and parameters All and Over, we will consider this setup:

Let k be the BaseField of A and k_1 the coefficient field of k. If All is true, let g := Aut(k/k_1), otherwise, g := < Over >. In both cases we define k_0 := Fix(K, g). In particular, if k is normal over the coefficient field k_1 then k_0 = k_1 and g is the full Galois group. In general g is not required to contain k_1 automorphisms, so that any subset of the Q automorphism group is valid as input. By construction, k is normal over k_0, and g acts on the ideals of k. In general however, g does not act on the ideal groups used to define A.

Subsections

• Predicates
• Constructions

Predicates

IsAbelian(A) : FldAb -> BoolElt

    All: BoolElt                        Default: false

    Over: [Map]                         Default: []

Returns true if and only if the abelian extension A is abelian over k_0.

IsNormal(A) : FldAb -> BoolElt

    All: BoolElt                        Default: false

    Over: [Map]                         Default: []

Returns true if and only if the abelian extension A is abelian over k_0. This tests whether the defining ideal group is a g-module.

IsCentral(A) : FldAb -> BoolElt

    All: BoolElt                        Default: false

    Over: [Map]                         Default: []

Returns true if and only if the abelian extension A is abelian over k_0. This tests whether the defining ideal group is a g--module with trivial action: If N is the norm group of A, the group extension 1 to N to G to g to 1 is central.

Constructions

GenusField(A): FldAb -> FldAb

    All: BoolElt                        Default: false

    Over: [Map]                         Default: []

The genus field is the maximal abelian extension of k_0 that is contained in the abelian extension A. The result of this function is an abelian extension of k_0.