Normal extension Can anybody help me with this problem ? Let $P\in {\mathbb Q}[X]$ be an irreducible polynomial of degree $p$ prime and let $z_1, z_2, \ldots ,z_p\in {\mathbb C}$ be the roots of $P$. Suppose that $z_2\in {\mathbb Q}(z_1)$. Prove that the extension ${\mathbb Q}(z_1)/{\mathbb Q}$ is normal.

Hints:
== The roots $\,z_1,...,z_p\,$ are different pairwise ;
== $\;[\Bbb Q(z_1):\Bbb Q]=p\;$ ;
== The embedding $\,\phi:\Bbb Q(z_1)\to\overline{\Bbb Q}\;$ determined by $\;\phi(z_1)=z_2\;$ is, in fact, a $\,\Bbb Q-$automorphism of $\,\Bbb Q(z_1)\,$ and thus $\,\left|\text{Gal}\left(\Bbb Q(z_1)/\Bbb Q\right)\right|\ge 2\;$
