Normal closure of finite extension

Question

Let $[E : F]$ be a finite extension, and let $N$ be the normal closure of $E$ over $F$. Let $\alpha \in E$ which does not belong to $F$. Then $N$ is a normal extension of $F(\alpha)$. But is it true, that $N$ is also a normal closure of $E$ over $F(\alpha)$?

I ask this question, because this is asserted in a step of the proof of Theorem 15.4 in the book Field Theory by Ian Adamson (Available here https://archive.org/details/IntroductionToFieldTheory/page/n53/mode/2up)

Answer 1

What do you get with $F=\Bbb{Q},E=\Bbb{Q}(2^{1/4}), \alpha= 2^{1/2}$