Is following a normal field extension?

Is $\mathbb{Q}(i)\in\mathbb{Q}(i)(\sqrt[4]{5})$ a normal field extension and if not give an extension of it which is normal over $\mathbb{Q}(i)$.

I saw $((i)(\sqrt[4]{5}))^2=-\sqrt[]{5}$ and I guess this is the minimal polynomial which has $(i\sqrt[4]{5})$ as root, but it is not a polynomial in $\mathbb{Q}(i)$ so I would say it is not normal and a new extension which might be normal might be $\mathbb{Q}(i+1)(\sqrt[4]{5})$. I know that this extension is normal over $\mathbb{Q}(i)(\sqrt[4]{5})$ but I am not sure if it follows also for $\mathbb{Q}(i)$.

Would be very happy for help.

Solution 1:

It is a normal extension. Note that the roots of the polynomial $x^4-5$ over $\mathbb Q(i)$ are exactly given by $\sqrt[4]5$, $i\sqrt[4]5$, $-\sqrt[4]5$ and $-i\sqrt[4]5$. These are all contained in $\mathbb Q(i,\sqrt[4]5)$.