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?

>>5
Maybe simpler to just say that Q is a field and C is algebraically closed? Or just that x^2+1 splits in C.

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