Files in this item
Files  Description  Format 

application/pdf 8410069.pdf (3MB)  (no description provided) 
Description
Title:  Genus Fields and Central Extensions of Number Fields 
Author(s):  Watt, Stephen Bruce 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  We study the (narrow) genus group of an abelian extension of number fields using a four term exact sequence of abelian groups derived from work of Frohlich. There are two main results. First, if L/K is a cyclic lextension, where l is a prime not dividing h(,K)('+), the narrow class number of K, then we determine the ltorsion subgroup of the genus group of L/K. Also, if K is imaginary quadratic then we determine all cyclic lextensions for which l (VBAR) h(,L). Next we consider central extensions. Let L/K be a finite Galois extension of number fields with Galois group (GAMMA) and let M((GAMMA)) = H(,2)((GAMMA), ). If E/K is a Galois extension which is central with respect to L/K then there is a canonical homomorphism from M((GAMMA)) into Gal(E/L). We say E realizes M((GAMMA)) if this homomorphism is injective. Now suppose K is imaginary quadratic and L/K an lextension such that the group of units of K has no ltorsion. We prove that M((GAMMA)) can be realized by a finite lextension E of K which is central with respect to L/K and has no additional ramification in the sense that a prime of K is ramified in E only if it is ramified in L. It follows that if (GAMMA) has lrank at least four then there is an infinite tower of finite lextensions of K containing L with no additional ramification. Now let S by any finite set of finite primes of K and K(l,S) the maximal lextension of K nonramified at the primes of K outside S. We use the result on realization of the multiplicator stated above to prove that a full set of defining relations for the prolgroup (OMEGA) = Gal(K(l,S)/K) can be lifted from the nontrivial abelian relations of the maximal abelian quotient group (OMEGA)('ab). We also exhibit specific generators and relations for (OMEGA)('ab) when l (VBAR) h(,K). 
Issue Date:  1983 
Type:  Text 
Description:  130 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1983. 
URI:  http://hdl.handle.net/2142/71217 
Other Identifier(s):  (UMI)AAI8410069 
Date Available in IDEALS:  20141216 
Date Deposited:  1983 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois