On Centralizer-Related Intermediate Fields

of a Skew Field Extension

 

Jan Treur

 

Vrije Universiteit Amsterdam, Department of Mathematics and Computer Science

De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Email: treur@cs.vu.nl   URL: http://www.cs.vu.nl/~treur

 

Abstract

For a skew field extension L/K, a number of intermediate fields can be defined related to the centralizer of K in L: based on an alternation of the constructions K.ZL(K) and ZL(ZL(K)). In this paper such intermediate fields are studied in detail. Among the results are a standard decomposition of any skew field extension of finite degree, and four types of skew field extensions that are basic for such a standard decomposition. A number of persistency properties of these types are explored. For three of the basic types it is shown how their structure can be described by structures of their centralizer. These three types are the only ocurring types in the case of finite [L : Z(L)]. It has been shown that K.ZL(K) and ZL(ZL(K)) are duals (cf. [7], [8]) of each other, and that the standard decomposition is self-dual. The special cases that one of these two intermediate fields is a subfield of the other one are discussed. Special cases of the results presented here can be related to Galois extensions and zeros of polynomials as discussed in [9], [10], and [11].

 

Keywords

skew field extension, intermediate field, centralizer, duality

 

References

 

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 9.     Treur, J., Separate zeros and Galois extensions of skew fields, Journal of Algebra, vol. 122 (1989), pp. 392-405

10.    Treur, J., Noncommutative splitting fields, Journal of Algebra, vol. 129 (1989), pp.367-379.

11.    Treur, J., Polynomial extensions of skew fields, Journal of Pure and Applied Algebra, vol. 67 (1990), pp. 73-93