Introduction

This version of Magma incorporates an interface to the KANT Version 4 system for algebraic function field computations. Special functions for rational function fields are described in Chapter RATIONAL FUNCTION FIELDS.

An algebraic function field F/k (in one variable) over a field k is a field extension F of k such that F is a finite field extension of k(x) for an element x in F which is transcendental over k. For perfect k it is always possible to choose x in F so that F/k(x) is also separable. For such x there exists a primitive element alpha in F with F = k(x, alpha) where alpha is a root of an irreducible, separable polynomial in k(x)[y].

Within Magma, user functions are provided for working with a finite extension F/k(x) or a transcendental extension F/k.

Algebraic function fields may be extended to create relative finite extensions of k(x) like F = k(x, alpha_1, ..., alpha_n). The functionality described in this chapter which is not available for these relative extensions is that involving series rings, galois groups and subfields.