Quotient of the Adele ring

Question

Greeting, i'd like to know why the quotient of the Adele ring ${\mathbb{A}}_\mathbb{Q}/\mathbb{Q}$ is compact and isomorphic to $\prod_p\mathbb{Z_p}\times \mathbb{R}/\mathbb{Z}$. Thanks in advance

Answer 1

You meant ${\mathbb{A_Q}}/i(\mathbb{Q})$ where $i$ is the diagonal embedding. Then it is immediate that for any $x\in {\mathbb{A_Q}}$ there is some $t\in \Bbb{Q}$ such that $x-i(t)\in \Bbb{\hat{Z}\times R}$ and hence $${\mathbb{A_Q}}/i(\mathbb{Q})=(\Bbb{\hat{Z}\times R})/(i(\mathbb{Q})\cap (\Bbb{\hat{Z}\times R}))=(\Bbb{\hat{Z}\times R})/i(\mathbb{Z})$$

It is a topological group because $i(\mathbb{Q})$ is discrete and it is compact because we have a continuous surjection from $\Bbb{\hat{Z}}\times [0,1]$ which is compact.