In class I learn that $\mathbb{Q}_3\not\cong\mathbb{Q}_5$ because one of them has $\sqrt{2}$ the other doesn't. Also professor asks us to find reference that $\mathbb{Q}_p\not\cong \mathbb{Q}_q\iff p\ne q$ but I can't find any. Could anyone provide me with a proof of this?
I'm familiar with inverse limit definition of $\mathbb{Z}_p$ and its field of quotient $\mathbb{Q}_p$, plus Power series definition of $\mathbb{Z}_p$ and Laurant series defintion of $\mathbb{Q}_p$ and further completion of $\mathbb{Q}$ with respect to p-adic norm. Feel free to use whichever you want. Thanks a lot.
This question must have come up many times. I gave an answer at Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes? , for instance.