abstract algebra question

Is there a shortcut to prove that Z, Q, R, S are pairwise nonisomorphic? I can split it into three proofs and do them individually, but I feel like there has to be some property that covers all the cases.

It can't be just the fact that Z < Q < R < S because the subset of even integers is isomorphic to the set of all integers.

Anyone care to push me in the right direction, if there is one?

>>6
(There really needs to be something here where we can edit/delete/preview posts amirite?)

Prove [math]\mathbb{Q}(\sqrt{2})[\math] and \mathbb{Q}(\sqrt{3}) are non-isomorphic.