How can I prove the following isomorphism?

Question

A profesor of mine said the following: Let $\mathbb{A}_{\mathbb{Q}}$ be the adele group of $\mathbb{Q}$. There exist a isomorphism of topological groups $$\frac{\mathbb{A}_{\mathbb{Q}}}{\mathbb{Q}}\simeq \varprojlim (\frac{\mathbb{R}}{n\mathbb{Z}})$$, where the limit of the right is considered with $\mathbb{N}$ ordered by divisibility. I tried to make a proof using that both spaces are compact and by trying to construct a continuous function. However, I have had no success. Do you know how to prove it or do you have any hint?

Answer 1

Hints:

1. Show $\displaystyle \varprojlim_n \mathbb{Z}/n\mathbb{Z}\cong \widehat{\mathbb{Z}}$ where $\widehat{\mathbb{Z}}:=\prod_p \mathbb{Z}_p$.
2. $\displaystyle \varprojlim_n \mathbb{R}/n\mathbb{Z}\cong \mathbb{Z}\setminus \mathbb{R} \times \varprojlim_n \mathbb{Z}/n\mathbb{Z}$ via noticing $\mathbb{R}/n\mathbb{Z}\cong \mathbb{R}/\mathbb{Z}\times \mathbb{Z}/n\mathbb{Z}$ canonically.
3. Show $\mathbb{Q}\setminus \mathbb{A}_{\mathbb{Q}}\cong \mathbb{Z}\setminus \mathbb{R}\times \widehat{\mathbb{Z}}$. For this, I think you need strong approximation property of $\mathbb{A}_{\mathbb{Q}}$ to get $\mathbb{A}_{\mathbb{Q}}=\mathbb{Q}+ \mathbb{Q}\times \prod_p \mathbb{Z}_p$, which suggests you how to define this isomorphism.