The Existence of Completely Free Elements

Abstract

In the present chapter we prove the existence of completely free elements in finite fields. For this purpose, in Section 14, we start considering the problem which arose in Chapter III, i.e., the existence of simultaneous generators for subgroups of (E, +) which carry various module structures arising from the intermediate fields of E over F (where E again is a finite dimensional cyclic Galois extension over F). In Section 14, we are concerned with simultaneous generators of submodules which are indecomposable with respect to the set of module structures considered. This leads to a fundamental problem which we have called the Two-Field-Problem. In Section 15 and Section 16 we study those situations where the Two-Field-Problem is easy to solve. In Section 15, when studying a local number theoretical condition, we are able to prove a result on so-called completely basic extensions. These are considered in Faith [12], Blessenohl [5], and Blessenohl and Johnsen [7]. In Section 17, we apply the results of the foregoing sections to extensions of prime power degree over finite fields, and complement the proof of Blessenohl and Johnsen’s Strengthening of the Normal Basis Theorem.