Document Type Master's Dissertation Author Van Zyl, Jacobus Visser 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 associatea 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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