Let $|\!\cdot\!|$ be a non-archimedean valuation on a field $K$. Let $V=\{x:|x|\leq 1\}$. Since $|\!\cdot\!|$ is ultrametric, $V$ is closed under addition and in fact an additive group. The other valuation axioms ensure that $V$ is a ring. We call $V$ the valuation ring of $K$ with respect to the valuation $|\!\cdot\!|$. Note that the field of fractions of $V$ is $K$.

The set $\mu=\{x:|x|<1\}$ is a maximal ideal of $V$. The factor $R:=V/\mu$ is called the residue field or the residue class field.

The map $\res:V\to V/\mu$ given by $x\mapsto x+\mu$ is called the residue map. We extend the definition of the residue map to sequences of elements from $V$, and hence to $V[X]$ so that if $f(X)\in V[X]$ is given by $\sum_{{i\leq n}}a_{{i}}X^{{i}}$ then $\res(f)\in R[X]$ is given by $\sum_{{i\leq n}}\res(a{i})X^{{i}}$.

Hensel property: Let $f(x)\in V[x]$. Suppose $\res(f)(x)$ has a simple root $e\in k$. Then $f(x)$ has a root $e^{{\prime}}\in V$ and $\res(e^{{\prime}})=e$.

Any valued field satisfying the Hensel property is called henselian. The completion of a non-archimedean valued field $K$ with respect to the valuation (cf. constructing the reals from the rationals as the completion with respect to the standard metric) is a henselian field.

Every non-archimedean valued field $K$ has a unique (up to isomorphism) smallest henselian field $K^{h}$ containing it. We call $K^{h}$ the henselisation of $K$.