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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Galois groups of maximal $ p$-extensions

Author: Roger Ware
Journal: Trans. Amer. Math. Soc. 333 (1992), 721-728
MSC: Primary 12F10; Secondary 12G05
MathSciNet review: 1061780
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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.

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Article copyright: © Copyright 1992 American Mathematical Society