Algebraic Function Fields

Within Magma, algebraic function fields of one variable can be created by adjoining a root of an irreducible, separable polynomial in k(x)[y] to the rational function field k(x). If k is a finite field, the function field is said to be global. An algebraic function field can be extended to create fields of the form $k(x, a_1, \ldots, a_r)$where each extension occurs by adjoining a root of an irreducible and separable polynomial. Not all the functionality below is present for extensions of algebraic function fields.

• Element arithmetic
• Norm, trace of an element with respect to the field extension F/k(x)
• Minimal and characteristic polynomials of an element wrt. F/k(x)
• Representation matrices of algebraic functions wrt. F/k(x)
• Construction of (`"finite'" and `"infinite'") equation orders
• Construction of (`"finite'" and `"infinite'") maximal orders (Round 2 algorithm)