The Genesis of a Theorem in the Galois Theory of <i>p</i>-Extensions of ℚ with Restricted Tame Ramification

Abstract

This article traces the genesis of a theorem that gives for the first time examples of the Galois group <I>G<SUB>S</SUB></I> of the maximal <i>p</i>-extension of ℚ, unramified outside a finite set of primes not containing an odd <i>p</i>, that are of cohomlogical dimension 2 if the primes in <I>S</I> satisfy a certain linking condition. Because the ramification is tame the pro-<i>p</i>-group <I>G<SUB>S</SUB></I> has all of its derived factors finite which is a strong finitenesss condition on <I>G<SUB>S</SUB></I>. The paper starts with a question of Serre on one relator pro-<i>p</i>-groups and then a detour to discrete groups where the notion of strong freeness for a sequence of homogeneous Lie elements is given and a criterion for strong freeness is established. These notions are then carried over to pro-<i>p</i>-groups where the linking condtion on the primes of <I>S</I> is translated into a cohomological criterion for a pro-p-group to have cohomological dimension 2. An analysis is given of the work of Koch where he gives a weaker criterion for a pro-p-group to have have cohomological dimension 2. A connecttion is made with this work of Koch and that of the author which would have been sufficient to prove the fact that <I>G<SUB>S</SUB></I> was of cohomological dimension 2 for certain sets <I>S</I> had it been applied to investigate whether the linking condition was true for certain sets <I>S</I>. It is not known if the cohomological dimension of <I>G<SUB>S</SUB></I> is 2 if <I>S</I> does not satisfy this linking condition.