I'm studing profinte groups. I'm using Wilson's book "Profinite Groups". Here the ring of $p$-adic integers $\Bbb Z_p$ is introduced as inverse limit of rings $\Bbb Z/p^n\Bbb Z$.
I'm searching for some lecture notes (available on the web if possibile) that explain the following things (I can't find them in the Wilson's book):
- $\Bbb Z_p$ is a PID (and other informations about its ideals)
- $\Bbb Z_p\simeq\operatorname{End}(C_{p^{\infty}})$ where $C_{p^{\infty}}$ is the $p$-Prufer group.
- This last one is the most important: the torsion part of the group of unit in $\Bbb Z_p$ is cyclic: $T((\Bbb Z_p)^{\times})\simeq C_{p-1}$.
Any suggested reading will be appreciated. Many thanks to all.