# Galois Groups and Greenberg's Conjecture

### by David C. Marshall

## Abstract

In this thesis we consider the structure of a certain infinite Galois group over K, the 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 the quantities g and s may be explicitly computed in terms
of the p-rank of the ideal class group of K.
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 possible rank r2+1.

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Last updated 6/27/00.