Galois groups of maximal $p$-extensions

Abstract:

Let $p$ be an odd prime and $F$ a field of characteristic different from $p$ containing a primitive $p$th root of unity. Assume that the Galois group $G$ of the maximal $p$-extension of $F$ has a finite normal series with abelian factor groups. Then the commutator subgroup of $G$ is abelian. Moreover, $G$ has a normal abelian subgroup with pro-cyclic factor group. If, in addition, $F$ contains a primitive ${p^2}$th root of unity then $G$ has generators ${\{ x,{y_i}\} _{i \in I}}$ with relations ${y_i}{y_j} = {y_j}{y_i}$ and $x{y_i}{x^{ - 1}} = y_i^{q + 1}$ where $q = 0$ or $q = {p^n}$ for some $n \geq 1$. This is used to calculate the cohomology ring of $G$, when $G$ has finite rank. The field $F$ is characterized in terms of the behavior of cyclic algebras (of degree $p$) over finite $p$-extensions.