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manuscripta mathematica

, Volume 151, Issue 3–4, pp 477–492 | Cite as

Intersective \(S_n\) polynomials with few irreducible factors

  • Daniela Bubboloni
  • Jack SonnEmail author
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Economics and ManagementUniversity of FirenzeFlorenceItaly
  2. 2.Department of MathematicsTechnion – Israel Institute of TechnologyHaifaIsrael

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