Originally Posted by

**AlexP** I am trying to show that $\displaystyle \mathbb{Q}(\sqrt{2})$ and $\displaystyle \mathbb{Q}(\sqrt{3})$ are not isomorphic.

My question is this... Isomorphisms preserve subfields (including, in this case, $\displaystyle \mathbb{Q}$), and I WANT to say something like '$\displaystyle \mathbb{Q}$ must be preserved so the mapping acts as the identity on $\displaystyle \mathbb{Q}$ but then it must act as the identity on the whole field, but then clearly it is not an isomorphism so $\displaystyle \mathbb{Q}(\sqrt{2})$ and $\displaystyle \mathbb{Q}(\sqrt{3})$ cannot be isomorphic.'

But, is any of this true, or even a useful way of thinking about it? Sadly, I suspect not...