Non-Galois cubic extensions Is there a necessary and sufficient condition for when a cubic extension of $\mathbb{Q}$ is not a Galois extension?

In a rather trivial way, a cubic extension $\,\Bbb Q(\alpha)/\Bbb Q\,$ is not Galois iff it is not normal iff the minimal polynomial of $\,\alpha\,$ in $\,\Bbb Q[x]\,\,,\,\,p(x)\,$ say, has only one root in $\,\Bbb Q(\alpha)\,$, namely $\,\alpha\,$ itself, iff the quadratic 
$$\frac{p(x)}{x-\alpha}\in\Bbb Q(\alpha)[x]$$
is irreducible there...
