I am new to the p-adic numbers (and also not too familiar with fields other than the reals, the rationals and $\mathbb{Z}/p$ ($p$ prime)). I want to know if the p-adic fields $\mathbb{Q}_p, \mathbb{Z}_p$ are characteristic 0 fields, or what are their characteristics? Actually, is $\mathbb{Z}_p$ a field at all?
(note that the only definition i have seen and somewhat understood so far of the p-adic number fields is that they are extensions of $\mathbb{Q}, \mathbb{Z}$ respectively (using Cauchy seq wrt the p-adic metric)).
Also what is the relationship between the p-adic numbers and characteristic $p$ fields?
Best Answer
The ring $\Bbb Z_p$ is not a field, since $p$ is not invertible (its valuation is $1$, while the valuation of any unit is $0$). The fraction field is $\Bbb Q_p$, it is an infinite extension of $\Bbb Q$ (it is a completion of $\Bbb Q$), so it has characteristic $0$.
We have $\Bbb Z_p \cong \Bbb Z[[t]] / (t-p)$, but $\Bbb Z_p$ is not isomorphic to $\Bbb F_p[[t]]$. Moreover, $\Bbb Z_p / p^n \Bbb Z_p \cong \Bbb Z / p^n \Bbb Z$ for any prime $p$ and any $n \geq 0$. This gives you some relations with the rings of characteristic $p$.
(Notice that there are local fields of characteristic $p$, namely the finite extensions of the field of Laurent series $\Bbb F_q(\!(t)\!)$, where $q$ being a power of $p$).