Galois Module Structure

If the base field k for class field constructions is normal with respect to some subfield k₀, i.e. k/k₀ 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 k₁ the coefficient field of k. If All is true, let g := Aut(k/k₁), otherwise, g := < Over >. In both cases we define k₀ := Fix(k, g). In particular, if k is normal over the coefficient field k₁ then k₀ = k₁ and g is the full Galois group.

In general g is not required to contain k₁ automorphisms, so that any subset of the Q automorphism group is valid as input. By construction, k is normal over k₀, and g acts on the ideals of k. In general however, g does not act on the ideal groups used to define A.

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 k₀.

IsNormal(A) : FldAb -> BoolElt

    All: BoolElt                        Default: false

    Over: [Map]                         Default: []

Returns true if and only if the abelian extension A is normal over k₀. 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 central over k₀. If k is cyclic over k₀ then this is equivalent to checking if A is abelian over k₀. 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 k₀ that is contained in the abelian extension A. The result of this function is an abelian extension of k₀.

H2_G_A(A) : FldAb -> ModTupRng

For A such that A is normal over Q with base field k that is normal too, compute the 2nd cohomology group of the Galois group of k acting on the ideal group defining A.

NormalSubfields(A) : FldAb -> []

    Quot: SeqEnum[RngIntElt]            Default: []

For an abelian extension, normal over Q and defined over a normal number field k as base field, return a list of all normal intermediate fields. If Quot is given, restrict to fields where the norm group has the abelian invariants as specified in Quot.

AbelianSubfield(A, U) : FldAb, GrpAb -> FldAb

FixedField(A, U) : FldAb, GrpAb -> FldAb

    IsNormal: BoolElt                   Default: false

For an abelian extension A with norm group map G to I for some finite abelian group G and a subgroup U<G, define the field corresponding to G/U, ie. the field fixed by U. If IsNormal is given then any cohomology information that is present is transferred to the new field - if possible.

CohomologyModule(A) : FldAb -> ModGrp, Map, Map, Map

For an abelian extension A defined over some normal field k/Q, compute the cohomology module (see Chapter COHOMOLOGY AND EXTENSIONS). The maps returned give the transition between the Z-modules used in the cohomology package and the ideal groups used to define A.

The first map returned maps between the automorphism group of k (as an permutation group) and the actual automorphisms of the field. It is obtained as the third return value of AutomorphismGroup.

The second map maps between the ideal group used to create A and a standart representation of the same group.

The third map maps between the standart representation of the norm group and the Z-module.