Lattices of Compatibly Embedded Finite Fields

The design of a computational facility for finite fields that allows complete freedom in the manner in which fields are constructed, is complicated by the fact that a field of fixed isomorphism typeKmay be constructed in many different ways. It is desirable that the user be able to perform simultaneous computations in different versions ofKin such a way that isomorphisms identifying elements in the different versions are applied automatically whenever necessary. This paper presents a coherent scheme for solving this problem based on an efficient method for compatibly embedding one field within another. This scheme forms a central component of the MAGMAmodule for finite fields. The paper also outlines the different representations of finite fields employed in the package and comments briefly on some of the major algorithms.