Splitting fields of G-varieties

Abstract

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X{)}^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K,G)\mapsto H^1(L,G)$ is trivial. If $L/K$ is a (finite) Galois extension then $\Gal(L/K)$ is called a splitting group of $X$. We prove a lower bound on the size of a splitting field of $X$ in terms of fixed points of nontoral abelian subgroups of $G$. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.

In this revision we simplified the proof of Lemma 4.3. AMS LaTeX 1.1, 36 pages. Author-supplied dvi file available at http://ucs.orst.edu/~reichstz/pub.html