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?

Seems to me it has to be at least three seperate ideas.

I'd start from the top down, obviously C isn't iso to any of the others because it has an element squaring to -1.

Next R isn't iso to Q or Z as R is uncountable and an isomorphism is a bijection

Q isn't isomorphic to Z because if it were, say isomorphism p:Q -> Z
p(1) = 1 by def. Also p(-1) = -1 and p(1+(-1))=0=p(1) + p(-1)
but then observe p(Z) = Z (excuse the notation), and thus the map isn't injective, as p(1/2) must also be in Z for example.

As isomorphism is an equivalence relation, this is sufficient to show pairwise non-isomorphic.

Is this simple enough? Strangely it's the last one that's the least neat, maybe there's a shorter proof than that.