Intersective $S_n$ polynomials with few irreducible factors

Abstract

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 $s(G)\leq r(G).$ 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 $r(S_n).$