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 AbstractTo every polynomialf(x) with rational coefficients one can associatea finite group

G_{f}, the Galois group of the splitting field offover the rational numbers. The inverse problem of Galois theory asks

whether for a given finite group

G, there exists a polynomialfsuchthat

Gis isomorphic toG_{f}. A Galois extension ofQ, with Galoisgroup

G, is called a realisation ofGoverQ, andGis said to occurover

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

GoverQexists if and only if a realisationexists over the function field

Q(x). Hence it suffices to findrealisations of a particular group

GoverQ(x), which enables us to usetools from Riemannian Surface Theory and Algebraic Geometry.

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