Find intermediate fields between $\mathbb{Q}$ and $\mathbb{Q}(2^{\frac{1}{4}}, i). $ Which of the extensions are normal over $\mathbb{Q}$?

Question

Find intermediate fields between $\mathbb{Q}$ and $\mathbb{Q}(2^{\frac{1}{4}}, i). $ Which of the extensions are normal over $\mathbb{Q}$ ?

I don't have any clue on how to procceed with this, I think I must find a polynomial for which $\mathbb{Q}(2^{\frac{1}{4}}, i). $ results to be its splitting field, but then I don't know what to do or how to determine which extension is normal, so any hint would be very appreciated.

Thank you!

Answer 1

Hint: $\mathbb{Q}(\sqrt[4]{2}, i)$ is the splitting field of $x^4-2$.