Abstract. It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois p-extension of a field F (where p is an odd prime) arise from p-henselian valuations with non-p-divisible value group, provided #(Ḟ /Ḟ p ) ≥ p 2 and F contains a primitive p-th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.

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We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincaré group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N ). Some other cases when G/N has virtually finite cohomological dimension are considered too.If G is profinite, the case of N projective or the profinite completion of the fundamental group of a compact surface is considered.

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