University of Illinois Urbana-Champaign

Genus Fields and Central Extensions of Number Fields

Watt, Stephen Bruce

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Description

Title
Genus Fields and Central Extensions of Number Fields
Author(s)
Watt, Stephen Bruce
Issue Date
1983
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(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 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).
Graduation Semester
1983
Type of Resource
text
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