Prove that $(\R,+)$ ({\mathbb R},+)$and$(\R[x],+)$({\mathbb R}[x],+)$ are isomorphic as abelian groups.
It is fairly easy to prove that they are actually isomorphic as $\Q$-vector {\mathbb Q}$-vector spaces, which is a stronger result; other than that I don't know any way of proving this. 1 [made Community Wiki] Here is another of my favourite examples: Prove that$(\R,+)$and$(\R[x],+)$are isomorphic as abelian groups. It is fairly easy to prove that they are actually isomorphic as$\Q\$-vector spaces, which is a stronger result; other than that I don't know any way of proving this.