# Intersective \(S_n\) polynomials with few irreducible factors

## Abstract

In this paper, an intersective polynomial is a monic polynomial in one variable with rational integer coefficients, with no rational root and having a root modulo *m* for all positive integers *m*. Let *G* be a finite noncyclic group and let *r*(*G*) be the smallest number of irreducible factors of an intersective polynomial with Galois group *G* over \(\mathbb {Q}\). Let *s*(*G*) be smallest number of proper subgroups of *G* having the property that the union of their conjugates is *G* and the intersection of all their conjugates is trivial. It is known that \(s(G)\le r(G)\). It is also known that if *G* is realizable as a Galois group over the rationals, then it is also realizable as the Galois group of an intersective polynomial. However it is not known, in general, whether there exists such a polynomial which is a product of the smallest feasible number *s*(*G*) of irreducible factors. In this paper, we study the case \(G=S_n\), the symmetric group on *n* letters. We prove that for every *n*, either \(r(S_n)=s(S_n)\) or \(r(S_n)=s(S_n)+1\) and that the optimal value \(s(S_n)\) is indeed attained for all odd *n* and for some even *n*. Moreover, we compute \(r(S_n)\) when *n* is the product of at most two odd primes and we give general upper and lower bounds for \(r(S_n)\).

## Mathematics Subject Classification

Primary 11R32 Galois theory## Preview

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