The purpose of this note is to extend Brumer's characterization of pro-pgroups of cohomological dimension two ([1], corollary 5·3) to presentations more general than free ones. The result is then used to rid a proof in [7] of certain field theoretic ingredients. As a by-product we complete a result of Tsvetkov [6] about group extensions obtained by omitting relations.
Abstract.Let the field F contain all /?-power roots of unity for some prime p and suppose that the absolute Galois group G of F is a one-relator pro-p group. We use Merkurjev-Suslin's theorem on the power norm residue symbol to show that G is an extension of a Demushkin group by a free pro-/) group.
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