I'm trying to solve a question which asks me to determine whether $\mathbb{Q}(\alpha)$ is a normal extension of $\mathbb{Q}$, where $\alpha^4 - 10\alpha^2 +1 =0.$
I know that to show an extension is normal it suffices to show it's the splitting field of some $f \in \mathbb{Q}[x],$ and the polynomial given has roots $ ± \sqrt{5 ± 2\sqrt6}$, so if I can show that $\mathbb{Q}(\sqrt{5 + 2\sqrt6}) = \mathbb{Q}(\sqrt{5 - 2\sqrt6})$ then not only does $\mathbb{Q}(\alpha)$ give the same field regardless of which root $\alpha$ is chosen, but also that $\mathbb{Q}(\alpha)$ must be the splitting field for $f = x^4 -10x^2 + 1 \in \mathbb{Q}[x]$ and is therefore normal.
Does anyone know of a good way to go about doing this (or, if I'm wrong and it's not normal how you'd go about showing that)?
Notice that $\sqrt{5+2\sqrt 6} = \frac 1{\sqrt{5- 2\sqrt 6}}$.