$p$-Adic Field

A $p$-adic field is a finite extension $k/\mathbb{Q}_p$ (hence a nonarchimedean local field).

Its ring of integers is $\mathcal O_k=\{x\in k: |x|_p\le 1\}$, with maximal ideal $\mathfrak p_k=\{x:|x|_p<1\}$.

A uniformizer is an element $\varpi\in\mathfrak p_k$ generating $\mathfrak p_k$.

The residue field is $\kappa_k:=\mathcal O_k/\mathfrak p_k$, a finite field of size $q$.