ON THE GALOIS STRUCTURE OF THE S-UNITS FOR CYCLOTOMIC EXTENSIONS OVER Q

Abstract

For K/k a finite Galois extension of number fields with G=Gal(K/k) and S a finite G-stable set of primes of K which is "large", Gruenberg and Weiss proved that the ZG-module structure of the S-units of K is completely determined up to stable isomorphism by: its torsion submodule, the set S, a particular character and the Chinburg class. In this Thesis, we will discuss the possibility of explicitly finding a ZG-module in the same stable isomorphism class of the S-units of K, in the particular case when k is the field of rational numbers and K is a cyclotomic extension over k.