具有较少不可约因子的交$S_n$多项式
Intersective $S_n$ polynomials with few irreducible factors
摘要：
交多项式是具有有理整数系数的一元多项式，没有有理根，并且对于所有正整数$m$都有一个根模$m$。设$G$是有限非循环群，$r(G)$是$\mathbb{Q}$上与Galois群$G$相交的多项式的最小不可约因子数.设$s(G)$是$G$的最小真子群数，其共轭的并是$G$，所有共轭的交是平凡的。已知$s(G)\leq r(G).$还已知如果$G$可实现为有理数上的伽罗瓦群，则它也可实现为交多项式的伽罗瓦群。然而，它是不知道的，在一般情况下，是否存在这样的多项式，这是一个产品的最小可行数$s(G)$的不可约因素。本文研究了$n$字母上的对称群$G=S_n$.我们证明，对于每个$n$，无论是$r(S_n)=s(S_n)$或$r(S_n)=s(S_n)+1$和最佳值$s(S_n)$确实达到所有奇数$n$和一些甚至$n$。当$n$是至多两个奇素数的乘积时，我们计算了$r(S_n)$，并给出了$r(S_n).$的一般上下界
英文摘要：
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 $r(S_n).$
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