Valuation Ring

Let $(K, \vert\cdot\vert)$ be a non-Archimedean valuated field. Its valuation ring $R$ is defined to be

$$R = \{ x \in K: \vert x\vert \leq 1\}.$$

The valuation ring has maximal Ideal

$$M = \{ x \in K: \vert x\vert <1\},$$

and the field $R/M$ is called the residue field, class field, or field of digits. For example, if $K = \Bbb{Q}_p$ ($p$-adic numbers), then $R = Z_p$ ($p$-adic integers), $M = pZ_p$ ($p$-adic integers congruent to 0 mod $p$), and $R/M$ = GF($p$), the Finite Field of order $p$.