Galois groups and Greenberg's conjecture

Abstract

We consider the structure of a certain infinite Galois group over Q(ζp) the cyclotomic field of p-th roots of unity. Namely, we consider the Galois group of the maximal p-ramified pro- p-extension. Very little is known about this group. It has a free pro-p presentation in terms of g generators and s relations where g and s may be explicitly computed in terms of the p-rank of the class group of Q(ζp). The structure of the relations in the Galois group are shown to be very closely related to the relations in a certain Iwasawa module. The main result of this dissertation shows this Iwasawa module to be torsion free for a large class of cyclotomic fields. The result is equivalent to verifying Greenberg's pseudo-null conjecture for the given class of fields. As one consequence, we provide a large class of examples of cyclotomic fields which do not admit free pro-p-extensions of maximal rank.