Determining whether $Q(i)(^4\sqrt 2) :Q(i)$ is a normal extension

Question

I'm trying to determine whether $\mathbb Q(i)(^4\sqrt 2) : \mathbb Q(i)$ is a normal extension

I have the polynomial $x^4 -2 \in \mathbb Q(i)$

Clearly all of its roots lie in $\mathbb Q(i)(^4\sqrt 2)$

So we have a splitting field, therefore the extension is normal.

Is that sufficient/correct?