M.Sc Student M.Sc Thesis Monderer Tal Genus 0 Subfields of Symmetric and Alternating Extensions Department of Mathematics ASSOCIATE PROF. Danny Neftin

Abstract

A traditional approach towards the study of polynomials is the investigation of their splitting fields. Given a bivariate polynomial f(x,t) with complex coefficient, it may also be viewed as a polynomial in x whose coefficients are rational functions over C. The splitting field of f when viewed in this way is a Galois extension F of the function field C(t). The Galois group of this extension may be viewed as a permutation group and thus investigated using tools from the theory of permutation groups. Since the base field was taken to be a function field, the theory of algebraic function fields introduces more machinery for describing the extension F/C(t) and its subfields. In particular, the correspondence between compact Riemann surfaces and function fields over C gives a notion of a genus for function fields over C.

Function fields of genus 0 and 1 are of interest due to some special properties: First, a function field over the complex numbers is rational if and only if it is of genus 0. Next, by Faltings' theorem, if a curve has infinitely many rational points, it must be of genus 0 or 1. Some questions on specializations of polynomials may be translated, using Faltings' theorem and basic properties of decomposition fields, into conditions on the genus 0 and 1 subfields of the splitting field of the polynomial.

In this work we consider the case where a Galois extension of C(t) has a symmetric or alternating Galois group G of large enough degree n and determine the subgroups of G fixing subfields of F of genus 0 or 1.

We show that a subgroup H that fixes a genus 0 or 1 subfield of a symmetric extension in general must lie between the alternating group on n-1 elements and the symmetric group on n elements, with some more added possibilities in the case of several exceptional types; and that  each of these possibilities for such H does in fact occur in some symmetric extension F/C(t).