Fields and Orders

Changes:

• FieldOfFractions returns a field of the new type FldOrd. Some of the functionality of number fields and their elements acting as field of fractions to orders has been transferred to this new type and is no longer present for number fields.

• The parameters Dedekind, Splitting and ReducedDiscriminant have all been removed from MaximalOrder and pMaximalOrder. Discriminant, Ramification and Al (for algorithm selection) parameters have been implemented for MaximalOrder.

• The i-th row of the j-th MultiplicationTable now contains the coefficients of the product of the ith and jth basis elements. This used to be the i-th column.

• The spelling of the option "Padic" for the Al parameter for GaloisGroup has been changed to "pAdic" to be consistent with spelling throughout the system. Fast was renamed into Conditional.

• General speed up of class group computations.

• NumberField(x^2-2) and NumberField(x^2-2) now return different fields.

• Improved root finder for embeddings of number fields and computing nth roots of algebraic numbers.

• The matrix returned from LLL has been transposed.

• Changed behaviour for creation of fields or orders with non-monic or linear polynomials.

• Order(RngOrd, AlgMatElt, RngIntElt), ideal<RngOrd| AlgMatElt> now have the matrix transposed.

• The names of all the Thue functions have changed. To create a thue object call Thue and to evaluate and solve call Evaluate and Solutions respectively.

• Relative norm equations will use a faster algorithm now. The new solver will find non-integral solutions for normal extensions.

New Features:

• The type FldOrd has been added. Fields of this type are the fields of fractions of an order. They will have the same basis as the order they are the field of fractions of which will not necessarily be a power basis. Elements of these fields have type FldOrdElt.

• The type FldAlg has been added as an overtype of FldNum and FldOrd and similarly FldAlgElt.

• Non monic polynomials can generate number fields.

• New varargs for number field construction: Abs, DoLinearExtension, Global to allow the creation of non-simple extensions, degree 1 extensions and ``global'' extensions.

• New signatures for basis functions which also take a ring as well as the order, field or ideal have been implemented returning the basis as elements of the given ring.

• Embed can be used to specify the embedding of one algebraic field in another and EmbeddingMap will return the existing embedding between two fields.

• Various properties of orders and their elements can be set by calling SetOrderMaximal, SetOrderTorsionUnit and SetOrderUnitsAreFundamental.

• The internal precision used by KANT for calculations in a real field can now also be set directly for fields (not just for orders).

• Solutions of IndexFormEquations can be calculated.

• Completion of absolute maximal orders at a finite prime

• pSelmerGroup of absolute maximal orders at a list of primes.

• Conductor of orders, Different of orders, elements and ideals.

• OptimizedRepresentation will return the homomorphism now, all number field homomorphisms allow for inverses.

• FactorBasis, Relations, ClassGroupCyclicFactorGenerators, RelationMatrix

• A parameter for Regulator allows access to the current value without first having to compute the fundamental units.

• New (faster) algorithms for factoring monic polynomials over (absolute) maximal orders, embedding of fields.

• Maximal printing for orders in Kash style. SetKantPrinting(true); to print order elements in Kash style.

• Random for RngOrd, FldAlg.

• Almost everything not dependent on real arithmetic (essentially class and unit groups) works for relative extensions (extensions of maximal orders).

• sub<...> and Order([ FldAlgElt ]) allow the construction of arbitrary orders.

• Use of the ``dot-operator .'' for orders and their fields of fractions to return the ith basis element as an element of the field.

• Galois theory: FixedField, FixedGroup. GaloisGroup and Subfields for (simple) relative extensions.

• Class field theory: a new type FldAb for the representation of Abelian extensions of number fields. Supporting functions: AbelianExtension, RayClassField, NormGroup, Discriminant, Degree, EquationOrder, Conductor, AbsoluteDegree, BaseField, BaseRing, CoefficientField, CoefficientRing, Components.

• AbsoluteDiscriminant, AbsoluteBasis.