Let $\val$ be a non-archimedean valuation on a field $K$ . Let $V=\{x:|x|\le 1\}$ . Since $\val$ 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 $\val$ . 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 $e\PR \in V$ and $\res(e\PR)=e$ .

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 $K$ .