Explicit integral Galois Module structure for low degree abelian extensions

Abstract

'Let M be a finite abelian extension of of Q. Then the ring of integers OM of M is a free, rank one module over the associated order AM/Q'. There is no method for constructing a Z(G) basis for OM when OM is free over Z(G). In this thesis Z(G) basis of OM over L is found whenever extension is tamely ramified, in the cases(i) when M, a bicyclic biquadratic extension of Q and L, quadratic subfield; (ii) M, a cyclic quartic, Galois extension of Q and L, quadratic subfield are considered for the explicit Galois Module Structure Problem. A Field extension M over L is considered where M is a quartic Galois extension of Q and L, it's quadratic subfield. The explicit structure of the associated order is given and conditions under which the ring of integers of M will be free over AM/L as a module and whenever it is free, a generator of OM over AM/L is given. Also the structure of OM as a Z(G) module is studied whenever M is tame over L. It is found explicitly the associated order and the structure of OM as an AM/L- for two different cases, when L = Q(w), G = Z3 where w is a primitive cube root of unity; L = Q(i),M = L [4 Sq.Rt(a)] where i^2 = -1, and a is an integer which is fourth power free. Chapter 4 considers the field extension F of K where K = Q(i) and G = Z4. An integral basis of F over K is found and with this the explicit structure of AF/K and of OF as an AF/K Module. In each of the cases the author gives generator of OF over AF/K.