It is obvious that $\mathbb{Q}_r$ is topologically isomorphic to $\mathbb{Q}_s$ where $r$ and $s$ denote different primes. But I really don't know whether it is true in the aspect of algebra. As I failed to prove it, I think that it is false, but I can't give a counterexample. Last I'm quite sorry that I'm new to MathJax and I don't know how to use it properly. Thanks for reading and I would appreciate it if you could solve my problem.