Separate Zeros and Galois Extensions of Skew Fields

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 [Z] 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 eiement 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: