Event № 153
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 $\dQ$. 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, even for the symmetric groups $S_n$, whether there exists such a polynomial which is a product of the smallest feasible number $s(G)$ of irreducible factors.
Theorem: For every $n$, either $r(S_n)=s(S_n)$ or $r(S_n)=s(S_n)+1$, with the first equality holding for all odd $n$. When $n$ is the product of at most two odd primes, $r(S_n)$ is computed explicitly. General upper and lower bounds for $r(S_n).$ are also given. (Joint work with Daniela Bubboloni)