Let p be prime with Show that there are integers a,b for which
By the other arguments is not irreducible. Thus, there exists such that take the norm of both sides to get . Now work with unique factorization over . It cannot be that one of those factors is and the other because we cannot write if . Thus, it means . (In fact, , and are exactly the same, i.e. there is also uniqueness involved but that is a different story).