Topics in Inverse Galois Theory

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.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2011

Author dc:creator

• Wills, Andrew Johan

Chair dc:contributor.committeechair

• Brown, Ezra A.

Committee members dc:contributor.committeemember

• Floyd, William J.
• Loehr, Nicholas A.

• Kronecker-Weber Theorem
• Rigid Groups
• Inverse Galois Theory

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Dc Identifier Other

etd-05032011-124510

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/32160