The following result may help to determine the Galois group of an irreducible polynomial.
Theorem : Let $G$ be a transitive subgroup of the symmetric group $S_n$ and p a prime number in $]\frac n2;n-2[$. If $G$ contains a p-cycle then G is either $S_n$ or $A_n$.
Does somebody have a comprehensive proof of this fact ? Using basic arguments, please. References are wellcome.