{"id":{"repo_id":"ohiolink","oai_identifier":"oai:etd.ohiolink.edu:osu1365594642"},"canonical_url":"https://search.dev.ndltd.org/etd/ohiolink/oai:etd.ohiolink.edu:osu1365594642","repository":{"repo_id":"ohiolink","name":"OhioLINK","base_url":"https://etd.ohiolink.edu/acprod/odb_etd/ws/oai/oai"},"display":{"title":"On the Galois module structure of the units and ray classes of a real abelian number field","abstract":"We study the Galois module structure of the ideal ray class group and the group of units of a real abelian number field. Specifically, we derive explicit annihilators of the ideal ray class groups in the vein of the classical Stickelberger theorems. This is made possible by generalizing a theorem of Rubin which in turn allows us to describe a relationship between the Galois module structure of certain explicit quotients of units and the Galois module structure of the ray class group. Along the way, we're compelled to study the Galois module structure of the p-adic completion of the units. We derive numerous conditions under which we may conclude that this module is cyclic some of which allow for p to divide the order of the Galois group. Under those conditions, we are able to relate the annihilators of the p-parts of various explicit quotients of units to annihilators of the p-parts of the ray class groups in many cases. This is a generalization of a theorem of Thaine.","abstract_html":"We study the Galois module structure of the ideal ray class group and the group of units of a real abelian number field. Specifically, we derive explicit annihilators of the ideal ray class groups in the vein of the classical Stickelberger theorems. This is made possible by generalizing a theorem of Rubin which in turn allows us to describe a relationship between the Galois module structure of certain explicit quotients of units and the Galois module structure of the ray class group. Along the way, we're compelled to study the Galois module structure of the p-adic completion of the units. We derive numerous conditions under which we may conclude that this module is cyclic some of which allow for p to divide the order of the Galois group. Under those conditions, we are able to relate the annihilators of the p-parts of various explicit quotients of units to annihilators of the p-parts of the ray class groups in many cases. This is a generalization of a theorem of Thaine.","abstract_has_math":false,"creators":["All, Timothy James"],"institution":"The Ohio State University","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Sinnott, Warren"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-07-23","date_published":"2013-07-23","updated_at":"2026-07-24T03:37:31Z","subjects":["Mathematics","real abelian number field","class group","Stickelberger","units"],"languages":["English"],"rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sinnott, Warren"]},{"key":"dc:creator","label":"Author","values":["All, Timothy James"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-07-23"]},{"key":"dc:publisher","label":"Institution","values":["The Ohio State University / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["The Ohio State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","real abelian number field","class group","Stickelberger","units"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. 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We derive numerous conditions under which we may conclude that this module is cyclic some of which allow for p to divide the order of the Galois group. Under those conditions, we are able to relate the annihilators of the p-parts of various explicit quotients of units to annihilators of the p-parts of the ray class groups in many cases. This is a generalization of a theorem of Thaine."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","p.83","393.2 KB"]},{"key":"dc:title","label":"Title","values":["On the Galois module structure of the units and ray classes of a real abelian number field"]}]}],"canonical_facts":{"dc:contributor":["Sinnott, Warren"],"dc:creator":["All, Timothy James"],"dc:date":["2013-07-23"],"dc:description":["We study the Galois module structure of the ideal ray class group and the group of units of a real abelian number field. Specifically, we derive explicit annihilators of the ideal ray class groups in the vein of the classical Stickelberger theorems. This is made possible by generalizing a theorem of Rubin which in turn allows us to describe a relationship between the Galois module structure of certain explicit quotients of units and the Galois module structure of the ray class group. Along the way, we're compelled to study the Galois module structure of the p-adic completion of the units. We derive numerous conditions under which we may conclude that this module is cyclic some of which allow for p to divide the order of the Galois group. Under those conditions, we are able to relate the annihilators of the p-parts of various explicit quotients of units to annihilators of the p-parts of the ray class groups in many cases. This is a generalization of a theorem of Thaine."],"dc:format":["application/pdf","p.83","393.2 KB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642"],"dc:language":["English"],"dc:publisher":["The Ohio State University / OhioLINK"],"dc:rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"dc:subject":["Mathematics","real abelian number field","class group","Stickelberger","units"],"dc:title":["On the Galois module structure of the units and ray classes of a real abelian number field"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["The Ohio State University"]},"updated_at":"2026-07-24T03:37:31Z"}