Is $\Bbb{Q}_p/\Bbb{Z}_p$ isomorphic to $\Bbb{Q}/\Bbb{Z}$ as a group ?
My try: I know $\Bbb{Q}/\Bbb{Z}\cong \bigoplus_p:prime \Bbb{Q}_p/\Bbb{Z}_p$ as group.
If I could prove $\bigoplus_p:prime \Bbb{Q}_p/\Bbb{Z}_p\cong \Bbb{Q}_p/\Bbb{Z}_p$, the proof ends but this maybe circular arguing.