manuscripta mathematica

, Volume 151, Issue 3–4, pp 477–492

# Intersective $$S_n$$ polynomials with few irreducible factors

• Daniela Bubboloni
• Jack Sonn
Article

## 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

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