Polynomials with roots mod \(p\) for all primes \(p\) (Q2711636)

A finite group \(G\) is said to be coverable, if there exist two proper subgroups such that \(G\) is the set-theoretic union of the \(G\)-conjugates of them. The authors prove that the symmetric group \(S_n\) is coverable iff \(3\leq n\leq 6\). As a consequence the following result holds: Let \(f(X)\) be an integer polynomial, which has a root modulo \(p\) for all primes \(p\). If \(f\) is a product of \(\ell\) irreducible factors, none of which is linear, then \(\ell\geq 2\). If \(\ell=2\) and the Galois group of \(f\) over the rationals is isomorphic to \(S_n\) for some \(n\), then \(3\leq n\leq 6\).