Algebraic Function Fields

Extensions of Algebraic Function Fields can be made and a range of functions can be applied to fields created as such.

Removals and Changes:

• Representation matrices of function field elements and elements of their orders have been transposed to be consistent with other Magma matrices.

• It has been ensured that the first argument to the Decomposition functions is always a ring.

New Features:

• The TateLichtenbaumPairing of two (non relative) divisors can be formed.

• A Parameterization of a function field, given a divisor to determine the pole, can be gained.

• Magma level printing has been implemented for fields, orders, elements, ideals, places and divisors.

• Extensions of algebraic function fields and their orders can be formed. Extensions of global algebraic function fields can only be formed by setting the optional parameter Check to false due to a missing test for polynomial irreducibility over global function fields.

• Maximal Orders, both finite and infinite, of extensions of algebraic function fields can be found.

• Most related structures of extensions can be found. The structures which cannot are ExactConstantField of a field and Reduce of an order.

• Invariants such as Characteristic, Degree, DefiningPolynomial, Basis and Discriminant can be determined for relative extensions and their orders.

• UnitRank of a finite maximal relative order of a global function field can be computed.

• Homomorphisms of all function fields can now be formed using the hom<|> constructor.

• Elements of relative extensions and their orders can be created, converted to a sequence and can have the usual arithmetic operators applied to them.

• Elements of extensions can be tested for equality and belonging to a certain ring or field. The predicates IsDivisibleBy, IsZero, IsOne, IsMinusOne, IsNilpotent, IsIdempotent, IsUnit, IsZeroDivisor, IsRegular, IsIrreducible and IsPrime can be tested on them.

• The Norm, Trace, Minimal and CharacteristicPolynomials and RepresentationMatrix of an element of an extension can be calculated. Numerator and Denominator with respect to a given order can also be determined.

• Elements of extensions can be made from and converted to a ProductRepresentation and expressed as a RationalFunction.

• Ideals of extensions can be created from a list of generators or as the product of an element and an order. Arithmetic is possible for these new ideals as well as tests for equality and IsOne, IsZero, IsIntegral and IsPrime.

• The intersection, GCD and LCM of two ideals of extensions can be found. The order the ideal is of can be reported and the Denominator calculated.

• The MultiplicatorRing, Minimum, Norm and Generators of an ideal of an extension can be found. A TwoElement presentation can also be retrieved. Relative ideals can be split into an integral ideal and a denominator using IntegralSplit.

• The Basis, BasisMatrix and TransformationMatrix of a relative ideal can be gained.

• Ideals of extensions can be factorized. Decomposition of prime ideals can be performed. The valuation of an ideal at a prime ideal and valuation of elements at a prime ideal can be calculated. The Ramification and Inertia Degrees of such ideals can be calculated. The ResidueClassField of an ideal can be returned.

• The Module and Relations functions can be applied to elements of extensions of algebraic function fields though there are restrictions as to the ring the resulting module is over.

• Factorization of polynomials over extensions of algebraic function fields can be accomplished.

Bug Fixes:

• Printing of elements has been tidied.

• A bug in Expand has been fixed.

• GCD and LCM of divisors was sometimes missing some places in the support of the result - this has been fixed.