On duality for skew field extensions

Abstract

In this paper a duality principle is formulated for statements about skew field extensions of finite (left or right) degree. A proof for this duality principle is given by constructing for every extension L/K of finite degree a dual extension LJK, . These dual extensions are constructed by embedding a given L/K in an inner Galois extension N/K. The Appendix shows that such an embedding can always be constructed, and introduces the notion of an inner closure N for L/K. In Section 1 some properties of inner Galois (or bicentral) extensions are mentioned, leading to the notion of a dual extension and the duality theorem. In Section 2 it is shown that the basic structures of LJK can be described by those of a dual extension L,/K,. A survey of this is the translation table at the end of Section 2. Based on these translations the general duality principle is formulated. In Section 3 we establish dual connections between a number of notions known in the literature. For instance, it appears that cyclic Galois exten- sions and binomial extensions are duals of each other. In Section 4 dual connections are used to prove in an easy way some results on cyclic Galois extensions, generalizing Amitsur’s results [l]. Compared to [S], in this paper in Section 1 the proofs are much shorter and generalized to the new notion of a predual extension. In Sections 2 and 4 derivations are also handled. Section 3 follows some parts of Chapter 7 of [S]. In Section 4 most of the material is new, although some special cases already were handled in [S]. The construction in the Appendix is new. We continue with some basic terminology. By a field we mean a skew field; we denote fields by K, L, N, D, E with or without subscripts. If Kc L