Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes?

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.

Solution 1:

Never. Looking at the number of roots of unity in your field suffices to distinguish all ${\mathbb{Q}}_p$ for odd values of $p$, because the number of roots of $1$ there is precisely $p-1$. It's different for the $2$-adic numbers, since they have two roots of unity, same as the $3$-adics. But the $2$-adics have a square root of $-7$ and the $3$-adics don't, whereas the $3$-adics have a square root of $10$ and the $2$-adics don't.