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

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.Z_{L}(K)
and Z_{L}(Z_{L}(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.Z_{L}(K)
and Z_{L}(Z_{L}(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].

skew field extension, intermediate field,
centralizer, duality

1. Albert, A.A., Structure of algebras.
Providence, A.M.S. coll. publ., 1961 (revised printing).

2. Albert, A.A., Tensor products of quaternion algebras. Proc. Amer. Math. Soc. 35 (1972), pp. 65-66.

3. Cohn, P.M., Quadratic extensions of skew fields. Proc. London Math. Soc. (3) 11 (1961), pp. 531-556.

4. Cohn, P.M., Skew Field Constructions. Cambridge University Press, 1977. Extended version: Skew Fields: Theory of General Division Rings, Cambridge University Press, 1995.

5. Jacobson, N., Structure of rings. Providence, A.M.S. coll. publ., 1968 (revised edition).

6. Risman, L.J., Zero divisors in tensor products of division algebras. Proc. Amer. Math. Soc. 51 (1975), pp. 35-36.

7. Treur, J., A duality for skew field extensions, Ph.D. Thesis, 1976, Utrecht University

8. Treur, J., On duality for skew field extensions, Journal of Algebra, vol. 119 (1988), pp. 1-22

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