Title page for ETD etd-05032011-124510


Type of Document Master's Thesis
Author Wills, Andrew Johan
Author's Email Address awillsa@vt.edu
URN etd-05032011-124510
Title Topics in Inverse Galois Theory
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Brown, Ezra A. Committee Chair
Floyd, William J. Committee Member
Loehr, Nicholas A. Committee Member
Keywords
  • Kronecker-Weber Theorem
  • Rigid Groups
  • Inverse Galois Theory
Date of Defense 2011-04-19
Availability unrestricted
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.
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