Finite fields: normal bases and completely free elements (Q2785547)

Let \(E\) be a finite-dimensional Galois extension of \(F\). The normal basis theorem states that every such extension contains an element \(w\) whose conjugates under the Galois group of \(E\) over \(F\) form an \(F\)-basis or normal basis of \(E\), and \(w\) is referred to as free in \(E\) over \(F\). It was recently shown that there always exist elements in \(E\) which are simultaneously free over every intermediate \(K\) of \(E\) over \(F\).NEWLINENEWLINENEWLINEThis work considers the characterization, enumeration and explicit construction of completely free elements in finite extensions of finite fields. Chapter I introduces the basic properties of extensions and free elements, concluding with a discussion of the special case of extensions of prime power degree and an outline of the monograph. In Chapter II the module structure of the finite dimensional cyclic Galois extension \(E\) over \(F\) is considered. The normal basis theorem and cyclicity of the group imply the additive group of \(E\) is a cyclic module over the principal ideal domain \(F[x]\) whose minimal polynomial is \(x^{|G|}-1\). The situation is extended in Chapter III to include the simultaneous module structures arising from intermediate fields. The existence of completely free elements in finite fields is then established. Explicit constructions of free and completely free elements are given in the final chapter. The monograph is an extended version of the Habilitationsschrift of the author at the University of Augsburg.