A lemma in my lecture notes states that if $K$ is a subfield of $\mathbb{C}$ and $\zeta=\exp(2\pi i/p)$ then $K(\zeta)/K$ is Galois. They proved it by arguing that the minimal polynomial of $\zeta$ divides $x^p-1$ which is a separable polynomial, and all its roots are powers of $\zeta$, hence $K(\zeta)/K$ is a separable normal extension, thus Galois.
I understand that to show that an extension is Galois it is sufficient to show that it is separable and normal, for which it suffices to show that the minimal polynomial of any element splits with distinct roots in the extension, but I don't see how we've proved that here. We've proven it for the particular element $\zeta$ but not for any other elements it seems. I'd appreciate it if someone could explain this.