Intersective \(S_n\) polynomials with few irreducible factors

详细信息    查看全文

• 作者：Daniela Bubboloni ; Jack Sonn
• 关键词：Mathematics Subject ClassificationPrimary 11R32 ; Galois theory
• 刊名：manuscripta mathematica
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：151
• 期：3-4
• 页码：477-492
• 全文大小：479 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebraic Geometry
  Topological Groups and Lie Groups
  Geometry
  Number Theory
  Calculus of Variations and Optimal Control
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1785
• 卷排序：151

文摘

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 \(r(S_n)\).