Finding whether a field extension is normal and/or separable

Question

I have the field extension $\frac{\mathbb{Q}(\sqrt[4]{-7})}{\mathbb{Q}(\sqrt{-7})}$

And i have to show whether it is normal and whether it is separable. I know that this extension is both separable and normal iff $L$ is the splitting field of a separable polynomial $ f \in \mathbb{Q}(\sqrt{-7})[x]$.

Say i take the separable polynomial $ f = x^4 + 7$, and show that $\mathbb{Q}(\sqrt[4]{-7})$ is not the splitting field of f over $\mathbb{Q}(\sqrt{-7})$, am i able to conclude that this extension is neither normal nor separable? Or does it mean that the extension is not normal AND separable, but could be one of them?

Answer 1

Statement :

Any extension of degree $2$ over a well behaved field is normal.....

Question :

• Why is the above statement true? (Can you guess how well behaved it should be)??

• How is this relevant to your question??