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?

>>9
5 is retarded though.

That fact that Z is not divisible doesn't imply it's not ring isomorphic to the rationals. He's not given any form of a proof, he's merely listed a difference between the two things, if they weren't different there'd be no hope of them not being ring isomorphic.

Hell, R has multiplicative elements of finite order, take -1 for example.

I think >>4 is the best (lol samefag, but still right).

>>10
To prove that, oh look, let's use 4's idea.
If there's an isomorphism p, take p(sqrt(2)), p(sqrt(2))^2 = p(2) = 2.p(1) = 2, so that implies p(sqrt(2)) [an element inside of Q adjoined root 3] squares to 2, contradiction.

I'll do >>11 in a bit if I can be arsed