M.Sc Thesis
M.Sc Student Rabayev Daniel Polynomials with Roots Mod n for All Positive Integers n Department of Mathematics Professor Jack Sonn

Abstract

We are interested in polynomials with integer coefficients that have roots mod n for all n, or equivalently, have roots in the p-adic numbers for all p. We consider only nontrivial polynomials, i.e., having no rational roots. It is known that such a polynomial cannot be irreducible, hence must be a product of two or more irreducible factors. Furthermore, the Galois group of such a polynomial is n-coverable, i.e., a union of n conjugacy classes of n proper subgroups, where n is the number of the irreducible factors of the polynomial. Such polynomials consisting of two irreducible factors are of special interest. In that case, it can be shown that among the symmetric and alternating groups, only Sn, where 2<n<7, and An where 3<n<9 can occur as Galois groups of such polynomials. We have found explicit polynomials for these groups except for A7 and A8. In general, a sufficient condition that a Galois extension k of the Rational numbers is the splitting field of such a polynomial is that all the decomposition groups are cyclic. This sufficient condition was used in all of the above cases except for A6, where a specific 2-covering may be used. In this work we will prove (by constructing explicit polynomials) that the groups S4, S5, S6, A4, A5, A6 are 2-coverable and can be realized as Galois groups of polynomials having a root in the p-adic numbers for all prime number p.