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 H¹(K,G)↦H¹(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.

Authors