Galois groups of prime degree polynomials with nonreal roots

In the process of computing the Galois group of a prime degree polynomial f(x) over Q we suggest a preliminary checking for the existence of non-real roots. If f(x) has non-real roots, then combining a 187-1 result of Jordan and the classification of transitive groups of prime degree which follows from CFSG we get that the Calois group of f(x) contains A(p) or is one of a short list. Let f(x) is an element of Q[x] be an irreducible polynomial of prime degree p >= 5 and r 2s be the number of non-real roots of f(x). We show that if s satisfies s (s log s + 2 log s + 3) <= p then Gal(f) = A(p), S-p.