M.Sc Thesis

M.Sc StudentAkkerman Yonathan
SubjectGalois Groups of Compositions of Random Polynomials
DepartmentDepartment of Mathematics
Supervisors ASSOCIATE PROF. Chen Meiri
Full Thesis textFull thesis text - English Version


In 1892 Hilbert proved his irreducibility theorem and deduced that the generic polynomial for Sn, that is, fA(x)=Anxn?0 has Galois group Sn over Q(A), where A=(A0,?,An) is a tuple of random variables. Since then, much effort was devoted to estimating the probability that fa has a large Galois group, when n is fixed and a=(a0,?,an) runs through a box
[-N,N]n. Recently, there is a rising interest in a different model, the bounded box model, where N is fixed and n grows. Under certain assumptions on the box, the Galois group of a random polynomial in this model was shown to be large, that is, An or Sn, by Bary-Soroker-Kozma, Bary-Soroker-Koukoulopoulos-Kozma, and conditionally on GRH by Breuillard-Varju. Nevertheless, very little is known about generic polynomials for other groups In the bounded box model.

In this work, we consider the composition of a generic polynomial with quadratic polynomials in the bounded box model. Our main result asserts that the Galois group of f(x2), for a random polynomial f in the bounded box model, contains a large transitive subgroup of the S2wrSn with probability tending to 1. To this end, we combine classical theorems from algebraic number theory with earlier results on random polynomials and reduce the question to a group theoretic one. In particular, as an intermediary step, we prove that four random elements drawn uniformly from S2wrSn invariably generate a large transitive subgroup with probability bounded away from 0. Finally, we believe that our methods can be refined in order to generalize our result to composition of two general random polynomials f and g.