This version of Magma is based on the KANT Version 4 system for algebraic function field computations. Special functions for rational function fields are described in Chapter FldFunRat.
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 ∈F which is transcendental over k. As a Magma object it is of type FldFun with elements of type FldFunElt. For perfect k it is always possible to choose x ∈F so that F/k(x) is also separable. For such x there exists a primitive element α ∈F with F = k(x, α) where α is a root of an irreducible, separable polynomial in k(x)[y].
All function fields in Magma can be represented as described above, i.e. as k(x, α) where x is transcendental and α is algebraic. This is the representation which has the most functionality.
Alternatively, one may wish to consider a function field as a combined transcendental and algebraic extension of its constant field, more like a curve. Such a field would be a quotient of k[x, y]. Fields in this representation do not have orders.
Algebraic function fields may be extended to create relative finite extensions of k(x) like F = k(x, α1, ..., αn). The functionality described in this chapter which is not available for these relative extensions is that involving series rings, galois groups and subfields.
It is also possible to make non--simple extensions where more than one root of a polynomial is added at each step by extending by several polynomials. These extensions have the same functionality as the relative finite extensions, except that primitive elements and some functions involving differentials are not available.
Function fields represented as finite extensions may have orders. Some orders will have a basis different to that of their function fields. Orders may have a field of fractions. A field of fractions of an order is isomorphic to the function field of the order, however its elements are represented with respect to the basis of the order. So there are 3 different types of rings covered in this chapter, function fields (FldFun), orders of function fields (RngFunOrd) and fields of fractions of orders of function fields (FldFunOrd). Each ring type has its own corresponding element type.