= Non-Archimedean valued field
{title2=$(K,|\cdot|),\quad|x+y|\le\max(|x|,|y|)$}
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 $R=\{x:|x|\le1\}$ 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.
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