Intersective $S_n$ polynomials with few irreducible factors

TL;DR

This paper investigates the minimal number of irreducible factors in intersective polynomials with symmetric group Galois groups, establishing bounds and exact values for specific cases, advancing understanding of Galois realizations.

Contribution

It proves that for all n, the minimal number of irreducible factors is either equal to or one more than the minimal number of subgroups covering the group, and computes exact values for certain n.

Findings

For all n, r(S_n) equals s(S_n) or s(S_n)+1.

s(S_n) is attained for all odd n and some even n.

Exact values of r(S_n) are computed when n is the product of up to two odd primes.

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 $r(S_n).$