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Several invariants of an abelian extension can easily be obtained from the
ideal groups without first computing defining equations for the field.
Let A be an abelian extension.
Based on the conductor-discriminant relation made explicit by
[Coh00, Section 3.5.2], the discriminant
of the class field A is computed. This does not involve the
computation of defining equations. The second return
value is the signature of the resulting field.
The absolute discriminant of A as a number field over Q.
Conductor(A) : FldAb -> RngOrdIdl, [RngIntElt]
Computes the conductor of the abelian extension A, i.e. the smallest
ideal and the smallest
set of infinite places that are necessary to define A. The algorithm used
is based on [Pau96], [HPP97].
The degree of the abelian extension A.
The degree of the abelian extension A over Q.
CoefficientField(A) : FldAb -> Fld
BaseField(A) : FldAb -> Fld
The base field of the abelian extension A, that is
FieldOfFractions(BaseRing(A)).
CoefficientRing(A) : FldAb -> Rng
The base ring of the abelian extension A, that is
the maximal order used to define the underlying ray class group.
The norm group (see the definition of norm group)
used to define the abelian extension A.
The decomposition field of the finite prime p in the abelian extension
A as an abelian (sub)extension.
The decomposition field of the place p in the abelian extension A
as an abelian extension.
DecompositionGroup(p, A) : RngOrdIdl, FldAb -> GrpAb
The decomposition group of the finite prime p in the abelian extension A.
The abelian group returned
is a subgroup of the norm group.
The decomposition group of the place p in the abelian extension A.
The abelian group returned
is a subgroup of the NormGroup.
The "type" of the decomposition of the finite prime ideal p in the
abelian extension A as a
sequence of pairs
< f, e > giving the degrees and the ramification indices.
The "type" of the decomposition of the place p in the abelian extension
A as a sequence of pairs
< f, e > giving the degrees and the ramification indices.
Normal: BoolElt Default: false
The "type" of the decomposition over Q of the prime number p in the
abelian extension A as a
sequence of pairs
< f, e > giving the degrees and the ramification indices.
If Normal is set to true then the algorithm assumes that the base field
of A is normal. This is used to speed up the computations.
Normal: BoolElt Default: false
Computes the decomposition type of all elements in l and returns them
as a multi-set. The list l must only contain objects for which
DecompositionType is defined. If Normal eq true then
the underlying DecompositionType function must be able to deal with it
too.
If Normal is set to true then the algorithm assumes that the base field
of the abelian extension A is normal. This is used to speed up the
computations.
Normal: BoolElt Default: false
Computes the decomposition type over Q in the abelian extension A
of all prime numbers a ≤p ≤b and returns them
as a multi set.
If Normal is set to true then the algorithm assumes that
the base field
of A is normal. This is used to speed up the computations.
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