  In the process of computing the Galois group of a prime degree polynomial
$f(x)$ over $\mathbb Q$ we suggest a preliminary checking for the existence of
non-real roots. If $f(x)$ has non-real roots, then combining a 1871 result of
Jordan and the classification of transitive groups of prime degree which
follows from CFSG we get that the Galois group of $f(x)$ contains $A_p$ or is
one of a short list. Let $f(x)\in \mathbb Q [x] $ be an irreducible polynomial
of prime degree $p \geq 5$ 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) \leq p $
then $Gal (f)= A_p, S_p$.
