  We compute the Galois group of the splitting field $F$ of any irreducible and
separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with
characteristic different from two. The proofs require to distinguish between
two cases: whether or not the cubic roots of unity belong to $K$. We also give
a criterion to determine whether a polynomial as $f(x)$ is irreducible, when
$F$ is a finite field. Moreover, at the end of the paper we also give a
complete list of all the possible subfields of $F$.
