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.
Original language | English |
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Pages (from-to) | 392-405 |
Number of pages | 14 |
Journal | Journal of algebra (Print) |
Volume | 122 |
Publication status | Published - 1989 |