## Abstract

There are a lot of arithmetic consequences if a Galois group of a number field is of cohomological dimension ≦ 2 (cf. [Schmidt, J. reine angew. Math. 596: 115–130, 2006], [Schmidt, Doc. Math. 12: 441–471, 2007], [Schmidt, J. reine angew. Math. 640: 203–235, 2010]). But with class field theory we only have an approximate description of the relators of such groups, which makes it difficult to determine the cohomological dimension. There are several criteria (cf. [Labute, Math. 596: 155–182, 2006], [Labute and Mináč, Mild pro-2-groups and 2-extensions of ℚ with restricted ramification, 2009]) on the so called linking numbers to get cd ≦ 2. The techniques in these papers use Lie algebra theory which become much more complicated for pro-2-groups. Here we will give a more simple and direct proof of the same algebraic criteria for a pro-*p*-group to be of cd ≦ 2 including the case *p* = 2.

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