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Here is another of my favourite examples: 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. |
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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. |
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