Some Results on Central Simple Algebras

Abstract

The present note is a continuation of the first part of [2].' The results of that part are applied here to obtain some new results on central simple algebras and some old ones in a new way. The first application is to the representation theory of the full linear group GL(n). It is shown that the degrees of the representation of GL(n) obtained by decomposing the mth power of an n-dimensional space are divisible by n/(n, m). The second application is to algebras of characteristic p 5 0. A new result obtained in this subject is that the isomorphism o7: a -> asp of a field e of characteristic p can be extended to a central simple algebra af over e if and only if v =_ 1 mod the exponent of Wf. The third application includes a new approach to the theory of cross products. In particular it is shown that the Brauer group of all division algebras split both by a field K and by a normal extension F of e whose Galois group is G is isomorphic with the first cohomology group