On Galois groups of prime degree polynomials with complex roots

Let f be an irreducible polynomial of prime degree p >= 5 over Q, with precisely k pairs of complex roots. Using a result of Jens Hochsmann (1999), we show that if p >= 4k + 1 then Gal(f / Q) is isomorphic to A(p) or S-p. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T. Shaska. If such a polynomial f is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree p over Q having complex roots.