Title page for ETD etd-12072005-121619


Document Type Master's Dissertation
Author Van Zyl, Jacobus Visser
Email koffie@tuks.co.za
URN etd-12072005-121619
Document Title Hilbert's irreducibility theorem and its application to the inverse Galois problem
Degree MSc (Mathematics)
Department Mathematics and Applied Mathematics
Supervisor
Advisor Name Title
Prof L M Pretorius
Keywords
  • hilbert's irreducibilty theorem
  • galois extensions
  • inverse galois problem
  • number theory
  • galois theory
Date 2005-11-30
Availability restricted
Abstract
To every polynomial f (x) with rational coefficients one can associate

a finite group Gf , the Galois group of the splitting field of f

over the rational numbers. The inverse problem of Galois theory asks

whether for a given finite group G, there exists a polynomial f such

that G is isomorphic to Gf. A Galois extension of Q, with Galois

group G, is called a realisation of G over Q, and G is said to occur

over Q. It is known that all abelian groups occur over Q, and Šaferevič showed in 1957 that all solvable groups occur over Q. Almost all other progress with the problem depends on Hilbert’s irreducibility theorem,

which implies that a realisation of G over Q exists if and only if a realisation

exists over the function field Q (x). Hence it suffices to find

realisations of a particular group G over Q (x), which enables us to use

tools from Riemannian Surface Theory and Algebraic Geometry.

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