On the index of cyclotomic units
This thesis is concerned with the unit group and class number of real abelian fields. We study subgroups of the unit group of such fields consisting of cyclotomic units and having index a multiple of the class number. We base our investigation on the work of Greither and Kučcera which provides a minimal generating set for such a subgroup. We refine this minimal generating set so as to obtain subgroup of the unit group whose index is a much smaller multiple of the class number. We outline a strategy for obtaining such refinements in general, and implement it in detail for conductors having two or three prime divisors. In the case of the conductor being divisible by two primes, our cyclotomic units generate a subgroup of index at most twice the class number. In the case of three primes, the index is a somewhat larger multiple of the class number whose exact shape depends on the ramification data.