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|Title:||Genus Fields and Central Extensions of Number Fields|
|Author(s):||Watt, Stephen Bruce|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|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 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).
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1983.
|Date Available in IDEALS:||2014-12-16|