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
$$\mathbb Z_p:=\varprojlim_n \mathbb Z/p^n\mathbb Z$$
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 $\mathbb Z_p$.