Algebraic Function Fields [HB 57]

The functionality of extensions of algebraic function fields has been extended to almost match that of ordinary algebraic function fields. The functionality of the orders of algebraic function fields has been extended to almost match the functionality of the orders of algebraic number fields.

Removals and Changes:

• The tuples returned by DecompositionType are now of the form $\langle$f, e$\rangle$ where f is the inertia degree and e is the ramification degree of the corresponding ideal or place in the decomposition. It is no longer always necessary to compute the decomposition to obtain this information.

• Factorization of ideals has been improved by storing a product representation on appropriate ideals and using a coprime factorization algorithm on this partial factorization.

• Eltseq of an element of an order is now over the field of fractions of the coefficient ring.

• Printing of divisors is no longer only done as a linear combination of places. In some cases this was too expensive.

New Features:

• Experimental introduction of class field theory for function fields. The new facilities include functions for RayClassGroup computation, defining equations of class fields and computation of norm groups.

• Places and Divisors of extensions of algebraic function fields can be created. They have the full functionality of the extensions of rational function fields except for the functions mentioned below.

• Differentials of extensions of algebraic function fields can be created. They have the full functionality of the extensions of rational function fields except for the functions mentioned below.

• All functions for algebraic function fields can be called on extensions of function fields with the exception of those involving series rings, galois groups and subfields. These functions are Reduce, Expand, Residue, GaloisGroup, Subfields, Automorphisms, IsSubfield and IsIsomorphic.

• The functions Modexp and Modinv can now be called on elements of orders of all algebraic function fields.

• RationalFunction can now be returned over a coefficient ring or field given as a second argument.

• The following intrinsics have been preexisting for orders of number fields and have recently been added for orders of algebraic function fields :
  • !! for ideals,
  • + for orders,
  • AbsoluteOrder, AbsoluteDiscriminant, Basis of an order of an algebraic function field over a ring given as a second argument and BasisMatrix of an order,
  • Different for orders, ideals and elements,
  • Index, EquationOrder, IsAbsoluteOrder for orders,
  • IsInert, IsRamified, IsSplit, IsTamelyRamified, IsTotallyRamified, IsTotallySplit, IsUnramified, IsWildlyRamified for ideals and orders,
  • creating an order from a basis,
  • PrimitiveElement, Simplify, SubOrder, pMaximalOrder, pRadical for orders,
  • ColonIdeal, meet of an ideal with a ring, IsPower and Root functions for ideals.

• The function RationalExtensionRepresentation now allows even relative extensions to be expressed as a direct extension of the rational function field. Expressing an algebraic function field as an extension of one of its coefficient fields can be accomplished using UnderlyingField.

• Completions of non relative function fields and their orders at ideals of degree one over the constant field (places of degree 1) can now be taken.

• ConstantFieldExtension extends a function field by extending the constant field.

• The following intrinsics were preexisting for extensions of rational function fields and are now available for extensions of algebraic function fields :
  • Differential, Differentiation, SeparatingElement, DifferentiationSequence,
  • DifferentialSpace, DifferentialBasis,
  • SpaceOfDifferentialsFirstKind (SpaceOfHolomorphicDifferentials),
  • BasisOfDifferentialsFirstKind (BasisOfHolomorphicDifferentials),
  • CartierRepresentation, HasseWittInvariant, ClassGroupPRank,
  • Identity of a differential space, IsCanonical,
  • Divisor, PrincipalDivisor,
  • RamificationDivisor , WronskianOrders, GapNumbers, WeierstrassPlaces,
  • DifferentDivisor, CanonicalDivisor, ComplementaryDivisor,
  • Identity of a divisor group,
  • Degree, Minimum, IsConstant, IsSeparating, Zeros, Poles,
  • SerreBound, IharaBound,
  • NumberOfPlacesOfDegreeOneBound, NumberOfPlacesOfDegreeOne,
  • NumberOfPlaces,
  • HasPlace, Places, RandomPlace, DivisorOfDegreeOne,
  • ClassGroupGenerationBound, ClassNumberApproximation,
  • LPolynomial, Genus, ExactConstantField,
  • DegreeOfExactConstantField (DimensionOfExactConstantField) .
  Some of these functions compute the rational extension representation of the relative field and perform the computations on this. Others can do the computations on the relative field directly.

• By means of a Type parameter to FunctionField rational function fields can be created as algebraic. Extensions of such fields will not be considered relative but still as rational extensions.

• The Trace and Norm of an element can be computed over a coefficient ring or field given as a second argument.

• The ideal constructor has been expanded to allow an ideal to be created from a basis and a denominator. The result is checked to see whether the input did indeed define a true ideal.

• Homomorphisms from orders of function fields can be created by giving the images of the basis elements.

• An extra argument has been added to Zeros, Poles and CommonZeros. The first argument can now be the field the places to be returned should be of.

Bug Fixes:

• A bug in Expand has been fixed.