Abstract. This paper is a working out of the same-titled talk given by the author at the Third International Conference on Finite Fields and Their Applications in Glasgow, 1995. We give a survey on recent results on the characterization, the structure, the enumeration, and the construction of completely free elements and normal bases in finite dimensional extensions over finite fields.

A Strengthening of the Normal Basis Theorem. If E is a finite dimensional Galois extension over a field F with Galois group G, then the Normal Basis Theorem states that the additive group (E, +) of E is a cyclic module over the group algebra FG, i.e., there exists an element w in E such that the set {g(w) | g ∈ G} of G-conjugates of w is an F-basis of E. Such a basis is called a normal basis in E over F. Every generator w of E as FG-module is called a normal basis generator in E over F. For the sake of simplicity such an element is also called free in E over F.

If H is a subgroup of G, and Fix(H) is the intermediate field of E over F belonging to H via the Galois correspondence, i.e., the subfield of E which is fixed elementwise by H, then (E, +) likewise carries the structure of a Fix(H) H-module.