A decomposition theory for cyclotomic modules under the complete point of view

In 1986, D. Blessenohl and K. Johnsen (1986, J. Algebra 103, 141-159) prove d that for any finite extension E/F of Galois fields there exists a complet e normal basis generator w of E/F, which means that w simultaneously genera tes a normal basis for E over every intermediate field of E/F. In a recent monograph by the author (1997, "Finite Fields: Normal Bases and Completely Free Elements," Kluwer Academic, Boston) a theory is developed which allows the study of module structures of Galois fields as extensions with respect to various subfields and which led to an exploration of the structure of c omplete normal basis generators as well as explicit and algorithmic constru ctions of these objects. In the present paper we continue the development o f that theory by providing various structural results: the Complete Decompo sition Theorem, the Complete Product Theorem, a Theorem on Simultaneous Gen erators, and a Uniqueness Theorem. (C) 2001 Academic Press.