THE FIRST TWO COHOMOLOGY GROUPS OF SOME GALOIS GROUPS

JAN MINAC ADRIAN WADSWORTH

Abstract. We investigate the first two Galois cohomology groups of p-extensions over a base field which does not necessarily contain a primitive pth root of unity. We use twisted coefficients in a systematic way. We describe field extensions which are classified by certain residue classes modulo p^{n}th powers of a related field, and we obtain transparent proofs and slight generalizations of some classical results of Albert. The potential application to the cyclicity question for division algebras of degree p is outlined.