I'm in the middle of a proof where I'd like to show that $\sqrt{2 - \sqrt{2}} \notin \mathbb{Q}(\sqrt{2 + \sqrt{2}})$
The only way I can think of involves finding an explicit set representation for $\mathbb{Q}(\sqrt{2 + \sqrt{2}})$.
At first I tried showing $\mathbb{Q}(\sqrt{2 + \sqrt{2}}) = \{a + b\sqrt{2 + \sqrt{2}}: a,b\in\mathbb{Q}\}$
and then realised this is probably false, as it doesn't look like it contains $\sqrt{2}$.
I figured I could try $\mathbb{Q}(\sqrt{2 + \sqrt{2}}) = \{a + b\sqrt{2 + \sqrt{2}} + c\sqrt{2}: a,b,c\in\mathbb{Q}\}$
but this method seems really long-winded. I'm pretty sure there'll be plenty of shorter methods, but I don't know any method to show this.
Any pointers appreciated!