{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/32160"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/32160","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Topics in Inverse Galois Theory","abstract":"Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q.","abstract_html":"Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q.","abstract_has_math":false,"creators":["Wills, Andrew Johan"],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Brown, Ezra A."],"committee_members":["Floyd, William J.","Loehr, Nicholas A."],"year":2011,"date_issued":"2011-04-19","date_published":"2011-04-19","updated_at":"2026-07-22T22:18:50Z","subjects":["Kronecker-Weber Theorem","Rigid Groups","Inverse Galois Theory"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-05032011-124510"],"render_values":[{"text":"etd-05032011-124510","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/32160","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Brown, Ezra A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Floyd, William J.","Loehr, Nicholas A."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Wills, Andrew Johan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:34:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:34:56Z","2011-05-19"]},{"key":"dc:date.issued","label":"Date","values":["2011-04-19"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Kronecker-Weber Theorem","Rigid Groups","Inverse Galois Theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-05032011-124510"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/32160"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:title","label":"Title","values":["Topics in Inverse Galois Theory"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Brown, Ezra A."],"dc:contributor.committeemember":["Floyd, William J.","Loehr, Nicholas A."],"dc:contributor.department":["Mathematics"],"dc:creator":["Wills, Andrew Johan"],"dc:date.accessioned":["2014-03-14T20:34:56Z"],"dc:date.available":["2014-03-14T20:34:56Z","2011-05-19"],"dc:date.issued":["2011-04-19"],"dc:description.abstract":["Galois theory, the study of the structure and symmetry of a polynomial or associated field extension, is a standard tool for showing the insolvability of a quintic equation by radicals. On the other hand, the Inverse Galois Problem, given a finite group G, find a finite extension of the rational field Q whose Galois group is G, is still an open problem. We give an introduction to the Inverse Galois Problem and compare some radically different approaches to finding an extension of Q that gives a desired Galois group. In particular, a proof of the Kronecker-Weber theorem, that any finite extension of Q with an abelian Galois group is contained in a cyclotomic extension, will be discussed using an approach relying on the study of ramified prime ideals. In contrast, a different method will be explored that defines rigid groups to be groups where a selection of conjugacy classes satisfies a series of specific properties. Under the right conditions, such a group is also guaranteed to be the Galois group of an extension of Q."],"dc:description.degree":["Master of Science"],"dc:identifier.other":["etd-05032011-124510"],"dc:identifier.uri":["http://hdl.handle.net/10919/32160"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Kronecker-Weber Theorem","Rigid Groups","Inverse Galois Theory"],"dc:title":["Topics in Inverse Galois Theory"],"dc:type":["Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:18:50Z"}