Find the smallest normal extension (up to isomorphism) of $\mathbb Q(\sqrt[4]2)$ in $\overline {\mathbb Q}$ (the algebraic closure of $\mathbb Q)$
My atempt:
$\mathbb Q(\sqrt[4]{2},i)$ is a normal extension of $\mathbb Q(\sqrt[4]{2})$, because $\mathbb Q(\sqrt[4]{2},i)$ is a splitting field of $x^4-2\in \mathbb Q(\sqrt[4]{2})$.
Am I right? this is indeed the smallest normal extension of $\mathbb Q(\sqrt[4]{2})$? I found difficult to prove this is the smallest normal extension $\mathbb Q(\sqrt[4]{2})$.
I need a hand here
Thanks