Polynomials which are locally reducible
  Let $K$ be a global field and $n > 1$ an integer. We show $n$ is composite if and only if there is an irreducible polynomial $f(x) \in K[x]$ of degree $n$ which is reducible $q$-adically for all the primes $q$ of $K$. 
R. Guralnick, M. Schacher, J. Sonn