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Watt, Stephen Bruce 2014-12-16T06:18:09Z 2014-12-16T06:18:09Z 10000-01-01 1983 1983 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 l-extension, where l is a prime not dividing h(,K)('+), the narrow class number of K, then we determine the l-torsion subgroup of the genus group of L/K. Also, if K is imaginary quadratic then we determine all cyclic l-extensions 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 l-extension such that the group of units of K has no l-torsion. We prove that M((GAMMA)) can be realized by a finite l-extension 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 l-rank at least four then there is an infinite tower of finite l-extensions 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 l-extension of K non-ramified 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 pro-l-group (OMEGA) = Gal(K(l,S)/K) can be lifted from the non-trivial 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). Made available in DSpace on 2014-12-16T06:18:09Z (GMT). No. of bitstreams: 1 8410069.pdf: 3110637 bytes, checksum: 606cfca5461fd75b928907c54b7531d0 (MD5) Previous issue date: 1983 Embargo set by: Seth Robbins for item 71383 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 130 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983. http://hdl.handle.net/2142/71217 (UMI)AAI8410069 Mathematics Genus Fields and Central Extensions of Number Fields text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.