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It is obvious that $\mathbb{Q}_r$ is topologically isomorphic to $\mathbb Q_s$ while $r$ 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.

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    What do you mean by "algebraically"? As rings? groups? vector spaces?2011-12-23
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    eh...It is a field.2011-12-23
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    Notice that if $p$ is another prime (so that $p$, $r$, $s$ are all different and say different from $2$) then $\mathbb{Q}_r$ contains $\sqrt{p}$ iff $p$ is a square mod $r$. And for any $p$ you can find $r$ and $s$ such that $(p/r)=1$ and $(p/s)=-1$.2011-12-23
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    Q3 is not isomorphic to Q5, for the following reason: any field isomorphism would have to map −1 to −1; but Q5 contains a square root of −1, whereas Q3 does not.Thanks to Martin Bright.2011-12-23
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    In fact, by Dirichlet's theorem (about primes in arithmetic sequences), for any $r$, $s$ you can find a prime $p$ such that $(p/r)=1$, $(p/s)=-1$ ($(./.)$ is the Legendre symbol). So it proves that all $\mathbb{Q}_r$'s are non-isomorphic as fields.2011-12-23
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    same question was posted in mathoverflow: http://mathoverflow.net/questions/84142/is-q-r-algebraically-isomorphic-to-q-s-while-r-and-s-denote-different-primes2011-12-23
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    And why are they topologically isomorphic? The metric thus defined are "different" in the sense that the topology they determined are distinct, is it not true?2011-12-23
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    Various answers are given (in an answer *stricto sensu*, and in the comments) in the MO version of the question, whose [link](http://mathoverflow.net/questions/84142/is-q-r-algebraically-isomorphic-to-q-s-while-r-and-s-denote-different-primes) has been given by @Paul.2011-12-23
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    Please define in the question the meaning of $Q_p$. This helps not only the reader, but the software platform, which has no tools to search based on mathematical notation.2011-12-23

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