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

This article traces the genesis of a theorem that gives for the first time examples of the Galois group GS of the maximal p-extension of ℚ, unramified outside a finite set of primes not containing an odd p, that are of cohomlogical dimension 2 if the primes in S satisfy a certain linking condition. Because the ramification is tame the pro-p-group GS has all of its derived factors finite which is a strong finitenesss condition on GS. The paper starts with a question of Serre on one relator pro-p-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-p-groups where the linking condtion on the primes of S 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 GS was of cohomological dimension 2 for certain sets S had it been applied to investigate whether the linking condition was true for certain sets S. It is not known if the cohomological dimension of GS is 2 if S does not satisfy this linking condition.