Algebraic Function Fields [HB 59]

In this release a new representation of algebraic function fields is provided -- non-simple extensions. Algebraic function fields can be created using more than one polynomial and still be a direct extension of the field which was being extended. Fields in this new representation have almost the same level of functionality as their representations as rational extensions. As in the case of the relative representations, Galois groups, series functions and subfields are not yet available. Some functions involving differentials also are not yet available.

Removals and Changes:

• The second argument to Completion is now a place rather than an ideal.

• For clarity the names of functions beginning NumberOfPlaces have had OverExactConstantField appended where applicable. Shorter synonyms have also been provided.

• The expansion of a function field element at a place has been improved to run substantially faster.

• Poles and Zeros of a function field element have been rewritten more efficiently.

• Element arithmetic has been made more efficient.

• Powering of elements in characteristic p rings and fields has been improved using the fact that (a + b)^p = a^p + b^p.

New Features:

• It is now possible to take non-simple extensions of function fields. Such extensions have all the functionality of simple extensions of function fields, except that functions involving series rings, galois groups, subfields and some functions involving differentials are not yet available.

• An order of a function field can be set to be either maximal or non-maximal using the intrinsic SetOrderMaximal.

• The calculation of Automorphism groups has been directly provided in AutomorphismGroup.

• A function field can be created from an existing function field by extending the constant field.

• Orders of function fields can be created by supplying an order and a sequence of elements of a function field. The order must be a maximal order of the coefficient ring or an order of the field containing the elements whose coefficient ring is maximal.

Bug Fixes:

• A bug in coercion of divisors into divisor groups over an extension of the original field has been fixed.