  We investigate the first two Galois cohomology groups of $p$-extensions over
a base field which does not necessarily contain a primitive $p$th 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.
