On rings of integers of relative abelian extensions of number fields

Abstract

By a number field, we mean a finite extension of Q. If F is a number field, the ring of integers of F is the integral closure of Z in F. Since the ring of integers of F is finitely generated and torsion free, it is free over Z. So Z-basis exists. To find the conditions for the existence of extension of number fields, and to compute it explicitly when it exists is an interesting problem in algebraic number theory. Mann's theorem is used in this thesis, for the discussions of problems in two particular types of integral bases. This thesis is dealing with the study of two problems (viz., Problem of Galois Module Structure, and a Problem with respect to Monogeneity), for two families of abelian extensions.