Non-Archimedean valued field ($(K,|\cdot |),|x+y|\le \max (|x|,|y|)$)

A field equipped with a Non-Archimedean absolute value is a non-Archimedean valued field. Its metric is $d(x,y)=|x-y|$ and satisfies the ultrametric inequality. The P-adic numbers give the basic mixed-characteristic examples. Completeness means that every Cauchy sequence for this metric has a limit in the field.

The sets $R=\{x:|x|\le 1\}$ and $\mathfrak{m}=\{x:|x|<1\}$ are its valuation ring and maximal ideal. Multiplicativity and the strong triangle inequality prove that $R$ is a ring and $\mathfrak{m}$ is an ideal. A nonzero element of $R$ is invertible in $R$ exactly when its absolute value is one, since $|x^{-1}|=|x{|}^{-1}$. Thus all nonunits are precisely $\mathfrak{m}$, proving that it is the unique maximal ideal and that $R/\mathfrak{m}$ is a field. These definitions concern a valuation, independently of whether the field admits an ordering.