Automorphisms

The group of relative automorphisms of the abelian extension is isomorphic via the Artin map to the ideal group used to define the field. After defining equations are computed, the user can explicitly map ideals that are coprime to the defining modulus to automorphisms of the field.

ArtinMap(A) : FldAb -> Map

Returns a map from the defining group (considered as a "subgroup" of the ideals of the base ring) into the automorphisms of the abelian extension A over the base field.

By the defining property of class fields, this map induces an isomorphism on the defining group (as an abelian group) onto the relative automorphisms of A.

Since this function constructs the number field defined by A, this may involve a lengthy calculation.

FrobeniusAutomosphism(A, p) : FldAb, RngOrdIdl -> Map

Computes the relative automorphism of the abelian extension A that is the Frobenius automorphism of p. Since this function constructs the number field defined by A, this may involve a lengthy calculation.

AutomorphismGroup(A) : FldAb -> GrpFP, [Map], Map

    All: BoolElt                        Default: false

    Over: [Map]                         Default: [ ]

If IsNormal is true for the abelian extension A with the given parameters, then the automorphism group of A over k_0 is computed. Since this function constructs the number field defined by A, this may involve a lengthy calculation.