  Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\QQ$,
with precisely $k$ pairs of complex roots. Using a result of Jens H\"{o}chsmann
(1999), we show that if $p\geq 4k+1$ then $\Gal(f/\QQ)$ 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 $\QQ$ having complex roots.
