Say:

$\displaystyle \alpha = \sqrt{2 + \sqrt{2}} $

And let:

$\displaystyle \beta = \sqrt{2 - \sqrt{2}} $

How do I show that $\displaystyle \beta $ is an element of the field extension:

$\displaystyle Q(\alpha):Q $

(the problem is to show that this field extension is normal, which I can do if I can show the above)

Any help with this would be appreciated, thanks.