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Algebraic Number Fields
Changes:
- Support for the use of complex approximations in the computation
of Galois groups over
Q has been added.
- A new algorithm to compute invariants for intransitive groups
yields a reduction of computation time of several orders
of magnitude for Galois groups of reducible polynomials.
- Removed an unnecessary restriction on orders being maximal when testing
whether ideals are prime.
- The MaximalOrder computation in Kummer extensions has been sped up
by avoiding the construction of intermediate p-maximal orders. (V2.15-3)
- The application of the residue field map to inputs with denominators has
been improved. (V2.15-6)
- Some expensive computations with orders have been avoided which has
considerable effect in maximal order computations. (V2.15-6)
- The factorisation of ideals is now returning an ordered sequence.
New Features:
- An implementation of Dirichlet and Hecke characters for number fields
has been added. Dirichlet characters are on elements of the field,
while Hecke characters are on ideals.
These are implemented as the dual group of RayResidueRing and
RayClassGroup respectively. The structures have been given
their Magma types, allowing substructures, pullbacks, etc., on the
groups of characters. A preliminary implementation of Hecke
Grössencharacters is also now available, at least for those
of type A0 over CM fields. This appears to be the first general
implementation for computing with Hecke Grössencharacters and their
L-functions.
- Computing a prime decomposition of a prime which totally ramifies in
a Kummer extension now uses a similar algorithm to that
implemented for computation of p-maximal and maximal orders.
- A new algorithm for the computation of the subgroup of K*
generated by a set of elements has been implemented. This allows
convenient access to the subgroups of the multiplicative group.
Bug Fixes:
- A problem with QuadraticClassGroupTwoPart in some cases was fixed.
- A bug in certain computations with trivial Ray class groups has been fixed.
Next: Algebraic Function Fields
Up: Arithmetic Fields (Global)
Previous: Dirichlet Characters
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