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?

>>25
R and C are isomorph as Q-module (As their base has the same cardinality, 2^aleph0). Every A-module morphism is a group morphism. So that imply that R and C are isomorph as groups. If you take R as an Q-module, you don't have the product between two reals, you have the structure of Q as a ring, and the structure of R as a group, and you define the "product" between elements of Q, with elements of R. In your post, you are considering R and C as fields, not as Q-vectorial spaces.