Separate zeros and Galois extensions of skew fields

Abstract

In the commutative case one has the following characterisation of Galois extensions of finite degree; we call L/K a Galois extension if K is the field of invariants of some group of automorphisms G of L: K= Inv G. (0) L/K is a Galois extension if and only if any polynomial p of degree m which is the minimal polynomial of an element of L has m distinct zeros in L. The notions separability and normality are related to this characterisation. In the case of skew fields polynomials often have infinitely many zeros, so a different way of counting zeros as distinct is needed. The well-known theorem of Gordon and Motzkin [2] states that a polynomial of degree n has zeros in at most n conjugacy classes. This suggests one should count zeros of a polynomial by the conjugacy classes in which they lie. However, in an inner Galois extension, for every minimal polynomial of an element all zeros are conjugates. That should count them as one. In this paper a different, more differentiated way of counting is proposed such that also in the case of an inner Galois extension the zeros of a polynomial p are counted as deg(p). In this paper we introduce a relation between zeros, called “separateness,” and count zeros by the maximal number of them which are separate. We prove that this notion has the following properties: (1) Any polynomial of degree m has at most m separate zeros. (2) If L/K is a Galois extension, then any polynomial p of degree m which is the (right) minimal polynomial of an element of L has m separate zeros Q1, . . . . Qm-1. The zeros in (2) can be taken uniform or of the same K-type; that is: for any i, j, Qi -> Qj, induces a K-isomorphism K(Qi) -> K(Qj). The converse of (2) holds in case L/K is a right polynomial extension, which means: there exists a generator Q such that 1, Q, . . . . Q^n-1 is a right basis of L/K. So for this type of extension a version of (0) holds.