{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71217"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71217","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Genus Fields and Central Extensions of Number Fields","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).","abstract_html":"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).","abstract_has_math":false,"creators":["Watt, Stephen Bruce"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:09Z","date_published":"2014-12-16T06:18:09Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8410069"],"render_values":[{"text":"(UMI)AAI8410069","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71217","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Watt, Stephen Bruce"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:09Z","10000-01-01","1983"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71217","(UMI)AAI8410069"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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."]},{"key":"dc:title","label":"Title","values":["Genus Fields and Central Extensions of Number Fields"]}]}],"canonical_facts":{"dc:creator":["Watt, Stephen Bruce"],"dc:date":["2014-12-16T06:18:09Z","10000-01-01","1983"],"dc:description":["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."],"dc:identifier":["http://hdl.handle.net/2142/71217","(UMI)AAI8410069"],"dc:subject":["Mathematics"],"dc:title":["Genus Fields and Central Extensions of Number Fields"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}