Intersective Sₙ polynomials with few irreducible factors

الأصل · EN

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 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)≤ 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=Sₙ, the symmetric group on n letters. We prove that for every n, either r(Sₙ)=s(Sₙ) or r(Sₙ)=s(Sₙ)+1 and that the optimal value s(Sₙ) is indeed attained for all odd n and for some even n. Moreover, we compute r(Sₙ) when n is the product of at most two odd primes and we give general upper and lower bounds for r(Sₙ).