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## Homework Statement

Let [tex]\beta[/tex] be a real, positive fourth root of 5. Is [tex]\mathbb{Q}(\sqrt{-5})[/tex] Galois over [tex]\mathbb{Q}[/tex]? How about [tex]\mathbb{Q}(\beta + i\beta)/\mathbb{Q}[/tex] or [tex]\mathbb{Q}(\beta + i\beta)/\mathbb{Q}(\sqrt{-5})[/tex]?

## Homework Equations

An extension is Galois when it is normal and separable.

## The Attempt at a Solution

Any push whatsoever in the right direction would be much appreciated.