{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71253"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71253","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"An Equal-Distribution Result for Galois Module Structure","abstract":"Let K be a number of field with ring of integers o, and let G be a fixed finite group. If K$\\sb\\pi$ is a tame Galois G-extension, the integral closure ${\\cal O}\\sb\\pi$ of o in K$\\sb\\pi$ is a locally free rank one oG-module, so realizes a class cl(${\\cal O}\\sb\\pi$) in the locally free class group Cl(oG). We let R(oG) denote the set of classes so realized. In the case where G is an elementary abelian group, we obtain the following result.","abstract_html":"Let K be a number of field with ring of integers o, and let G be a fixed finite group. If K<span class=\"etd-inline-math\">\\sb&pi;</span> is a tame Galois G-extension, the integral closure <span class=\"etd-inline-math\">{\\cal O}\\sb&pi;</span> of o in K<span class=\"etd-inline-math\">\\sb&pi;</span> is a locally free rank one oG-module, so realizes a class cl(<span class=\"etd-inline-math\">{\\cal O}\\sb&pi;</span>) in the locally free class group Cl(oG). We let R(oG) denote the set of classes so realized. In the case where G is an elementary abelian group, we obtain the following result.","abstract_has_math":true,"creators":["Foster, Kurt Christopher"],"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":"2014-12-16T06:18:18Z","date_published":"2014-12-16T06:18:18Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8721636"],"render_values":[{"text":"(UMI)AAI8721636","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71253","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Foster, Kurt Christopher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:18Z","10000-01-01","1987"]},{"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/2142/71253","(UMI)AAI8721636"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let K be a number of field with ring of integers o, and let G be a fixed finite group. If K$\\sb\\pi$ is a tame Galois G-extension, the integral closure ${\\cal O}\\sb\\pi$ of o in K$\\sb\\pi$ is a locally free rank one oG-module, so realizes a class cl(${\\cal O}\\sb\\pi$) in the locally free class group Cl(oG). We let R(oG) denote the set of classes so realized. In the case where G is an elementary abelian group, we obtain the following result.","Theorem. Let K be a number field with ring of integers o, and G an elementary abelian group. Let c $\\in$ R(oG), and denote by N(c,X) the number of tame Galois G-extensions K$\\sb\\pi$ for which cl(${\\cal O}\\sb\\pi$) = c and having absolute discriminant ${\\rm d}({\\cal O}\\sb\\pi/\\doubz) \\le {\\rm X}.$ Then N(c,X) $\\sim\\beta$ $\\cdot$ Y(log Y)$\\sp{\\rm r-1}$ where Y$\\sp{\\varphi(\\vert{\\rm G}\\vert)}$ $\\cdot$ d(o/$\\doubz$)$\\sp{\\vert{\\rm G}\\vert}$ = X. Here, $\\beta$ is a positive constant depending on K and G, but not on the class c $\\in$ R(oG), and r is a positive integer which depends only on K and G.","This is proved in Theorem (4.1). It tells us that the number of tame Galois G-extensions K$\\sb\\pi$ for which d(${\\cal O}\\sb\\pi/\\doubz) \\le$ X and cl(${\\cal O}\\sb\\pi$) = c $\\in$ R(oG) is asymptotically the same for each c $\\in$ R(oG). This is the equal distribution result of the title.","A result for fields is retrieved by noting that, for a tame Galois field extension L/K with Gal(L/K) = $\\Gamma$ isomorphic to G, a choice of isomorphism $\\Gamma \\cong {\\rm G}$ makes L into a G-extension. This is described in Chapter 4, and the result given by Theorem (4.2).","Made available in DSpace on 2014-12-16T06:18:18Z (GMT). No. of bitstreams: 1 8721636.pdf: 1854483 bytes, checksum: 87ddc2111a1f9973a3054830e60f99d8 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71419 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","68 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."]},{"key":"dc:title","label":"Title","values":["An Equal-Distribution Result for Galois Module Structure"]}]}],"canonical_facts":{"dc:creator":["Foster, Kurt Christopher"],"dc:date":["2014-12-16T06:18:18Z","10000-01-01","1987"],"dc:description":["Let K be a number of field with ring of integers o, and let G be a fixed finite group. If K$\\sb\\pi$ is a tame Galois G-extension, the integral closure ${\\cal O}\\sb\\pi$ of o in K$\\sb\\pi$ is a locally free rank one oG-module, so realizes a class cl(${\\cal O}\\sb\\pi$) in the locally free class group Cl(oG). We let R(oG) denote the set of classes so realized. In the case where G is an elementary abelian group, we obtain the following result.","Theorem. Let K be a number field with ring of integers o, and G an elementary abelian group. Let c $\\in$ R(oG), and denote by N(c,X) the number of tame Galois G-extensions K$\\sb\\pi$ for which cl(${\\cal O}\\sb\\pi$) = c and having absolute discriminant ${\\rm d}({\\cal O}\\sb\\pi/\\doubz) \\le {\\rm X}.$ Then N(c,X) $\\sim\\beta$ $\\cdot$ Y(log Y)$\\sp{\\rm r-1}$ where Y$\\sp{\\varphi(\\vert{\\rm G}\\vert)}$ $\\cdot$ d(o/$\\doubz$)$\\sp{\\vert{\\rm G}\\vert}$ = X. Here, $\\beta$ is a positive constant depending on K and G, but not on the class c $\\in$ R(oG), and r is a positive integer which depends only on K and G.","This is proved in Theorem (4.1). It tells us that the number of tame Galois G-extensions K$\\sb\\pi$ for which d(${\\cal O}\\sb\\pi/\\doubz) \\le$ X and cl(${\\cal O}\\sb\\pi$) = c $\\in$ R(oG) is asymptotically the same for each c $\\in$ R(oG). This is the equal distribution result of the title.","A result for fields is retrieved by noting that, for a tame Galois field extension L/K with Gal(L/K) = $\\Gamma$ isomorphic to G, a choice of isomorphism $\\Gamma \\cong {\\rm G}$ makes L into a G-extension. This is described in Chapter 4, and the result given by Theorem (4.2).","Made available in DSpace on 2014-12-16T06:18:18Z (GMT). No. of bitstreams: 1 8721636.pdf: 1854483 bytes, checksum: 87ddc2111a1f9973a3054830e60f99d8 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71419 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","68 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."],"dc:identifier":["http://hdl.handle.net/2142/71253","(UMI)AAI8721636"],"dc:subject":["Mathematics"],"dc:title":["An Equal-Distribution Result for Galois Module Structure"],"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"}