1
$\begingroup$

Q: If $p,q$ are different primes, show the embedding $$\mathbb{Q} \rightarrow \mathbb{Q}_q \times \mathbb{Q}_p$$ $$x \rightarrow (x,x)$$ is dense in the product space $\mathbb{Q}_q \times \mathbb{Q}_p$.

Furthermore, show the embedding $$\mathbb{Q} \rightarrow \mathbb{R} \times \mathbb{Q}_p$$ $$x \rightarrow (x,x)$$ is dense in the product space $\mathbb{R} \times \mathbb{Q}_p$.

My idea: The professor gives us a hint to use Chinese Remainder Theorem. However, I still have no idea to start off my solutions...Can anyone help to explain how to do this proof? I am confused about this topic also. Thank you.

$\endgroup$
  • $\begingroup$ That's two questions! $\endgroup$ – Lord Shark the Unknown Oct 20 '18 at 4:10
  • $\begingroup$ They are not the same? It seems they are similar to each other and I think the method is more or less the same... $\endgroup$ – Jason Ng Oct 20 '18 at 6:32
  • $\begingroup$ But then how should I start to show the first one?? $\endgroup$ – Jason Ng Oct 20 '18 at 10:30
1
$\begingroup$

Let's look at the second one. Given $a\in \Bbb R$, $b\in\Bbb Q_p$ and $\newcommand{\ep}{\varepsilon}\ep>0$ one needs to prove the existence of $x\in\Bbb Q$ with $|a-x|_\infty<\ep$ and $|b-x|_p<\ep$. There are certainly $a'$, $b'\in\Bbb Q$ with $|a-a'|_\infty<\ep/2$ and $|b-b'|_p<\ep/2$ so it suffices to prove that there is $x\in\Bbb Q$ with $|x-a'|_\infty<\ep/2$ and $|x-b'|_p<\ep/2$. Let $y=x-b'$ and $c=a'-b'$. Then we want $y\in\Bbb Q$ with $|y-c|_\infty<\ep$ and $|y|_p<\ep/2$.

To achieve this, consider $u=p^N/(1+p^{2N})$ where $N$ is a sufficiently large integer. Choosing $N$ large enough, gives $|u|_p<\ep/4$ and $|u|_\infty<\ep/4$ say. Then $|mu|_p<\ep/4$ for all integers $m$. We can now take $m\in\Bbb Z$ so that $mu$ lies in the interval $(c-\ep/4,c+\ep/4)$ inside $\Bbb R$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.