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 \(g := {\operatorname{Aut}}(k/k_1)\), otherwise, \(g := \langle \hbox{\tt Over}\rangle\). In both cases we define \(k_0 := {\operatorname{Fix}}(k, g)\). 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 \({\mathbb{Q}}\) 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 \(A\).

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 normal 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 central over \(k_0\). If \(k\) is cyclic over \(k_0\) then this is equivalent to checking if \(A\) 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 \(k_0\).

H2_G_A(A): FldAb -> ModTupRng

For \(A\) such that \(A\) is normal over \({\mathbb{Q}}\) 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.