Galois cohomology of fields without roots of unity

Abstract

There is a standard correspondence between elements of the cohomology group H 1( F, μ n ) (with the trivial action of Γ F =Gal( F s/ F) on μ n ) and cyclic extensions of dimension n over F. We extend this to a correspondence between the cohomology groups H 1( F, μ n ) where the action of Γ F on μ n varies, and the extensions of dimension n of K which are Galois over F, where K= F[ μ n ] and [ K: F] is prime to n. The cohomology groups are also related to eigenspaces of H 1(K, Z/n) with respect to the natural action of Gal( K/ F). As a result, we extend Albert's cyclicity criterion, stated in the 1930s for division algebras of prime degree, to algebras of prime-power degree over F, under the assumption stated above. We also extend the Rosset–Tate result on the corestriction of cyclic algebras in the presence of roots of unity, to extensions in which roots of unity live in an extension of dimension ⩽3 over the base field. In particular, if roots of unity live in a quadratic extension of the base field, then corestriction of a cyclic algebra along a quadratic extension is similar to a product of two cyclic algebras. Another application is that F-central algebras which are split by a certain semidirect product extension of F, are cyclic. In particular, if [ K: F]=2 then algebras over F which are split by an odd order dihedral extension, are cyclic. We also construct generic examples of algebras which become cyclic after extending scalars by roots of unity, and show the existence of elements for which most powers have reduced trace zero.