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.
|