Is a finite simple extension of fields in characteristic zero already normal?

Question

Let $k(\alpha) / k$ be a finite separable simple extension, $char(k) = 0$. Is $k(\alpha) / k$ already a normal extension? I can't come up with a counterexample or a proof that it is normal.

Answer 1

$\Bbb Q[\sqrt[3]{2}]$ (where we've adjoined the real cube root of two) is separable because everything in characteristic zero is separable. But it's not normal because it's not Galois (in general separable and normal implies Galois) because the other two roots of $x^3-2$ are not real so clearly are not contained in $\Bbb Q[\sqrt[3]{2}]$