Htykuut

In the following diagram (from the first edition):

I have a question about notation. Is $hat{mathbb{Z}}_{(p)}$ the ring of $p$-adic integers, that I have seen elsewhere denoted by $mathbb{Z}_p$?

What about $hat{mathbb{Q}}_p$? Is that the field of $p$-adic numbers, namely the metric completion of $mathbb{Q}$ with respect to the $p$-adic distance? I have seen this elsewhere denoted by $mathbb{Q}_p$ (assuming I am right).

As to $mathbb{C}_p$ is that the algebraically closed field which is obtained by first taking the algebraic closure of the field of $p$-adic numbers, and then its metric completion, for the extension of the $p$-adic metric?

I understand that notation changes from book to book, and that it also changes over time. Can someone please confirm if I understand the notation correctly? If not, can someone please correct me then? Thank you.

Yes, everything you're saying is correct. A more algebraic way of defining the p-adic numbers is as the inverse limit

$$mathbb Z_p:=varprojlim_n mathbb Z/p^nmathbb Z$$

which, alternatively, is the same as the completion of the local ring $mathbb Z_{(p)}$ [the localization of $mathbb Z$ at the prime ideal $(p)$] at its maximal ideal, hence the notation $hat{mathbb{Z}}_{(p)}$. Then $mathbb Q_p$ can be defined as the fraction field of $mathbb Z_p$.