Topics in Inverse Galois Theory

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dc.contributor.authorWills, Andrew Johanen
dc.contributor.committeechairBrown, Ezra A.en
dc.contributor.committeememberFloyd, William J.en
dc.contributor.committeememberLoehr, Nicholas A.en
dc.contributor.departmentMathematicsen
dc.date.accessioned2014-03-14T20:34:56Zen
dc.date.adate2011-05-19en
dc.date.available2014-03-14T20:34:56Zen
dc.date.issued2011-04-19en
dc.date.rdate2011-05-19en
dc.date.sdate2011-05-03en
dc.description.abstractGalois 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.en
dc.description.degreeMaster of Scienceen
dc.identifier.otheretd-05032011-124510en
dc.identifier.sourceurlhttp://scholar.lib.vt.edu/theses/available/etd-05032011-124510/en
dc.identifier.urihttp://hdl.handle.net/10919/32160en
dc.publisherVirginia Techen
dc.relation.haspartWills_AJ_T_2011.pdfen
dc.rightsIn Copyrighten
dc.rights.urihttp://rightsstatements.org/vocab/InC/1.0/en
dc.subjectKronecker-Weber Theoremen
dc.subjectRigid Groupsen
dc.subjectInverse Galois Theoryen
dc.titleTopics in Inverse Galois Theoryen
dc.typeThesisen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorVirginia Polytechnic Institute and State Universityen
thesis.degree.levelmastersen
thesis.degree.nameMaster of Scienceen

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