henselian field

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

The set is a maximal ideal of $V$. The factor$R:=V/\mu$ is called the residue field or the residueclass field.

The map $\mathrm{res}:V\to V/\mu$ given by $x\mapsto x+\mu$ is called theresidue map. We extend the definition of the residue map tosequences of elements from $V$, and hence to $V[X]$ so that if $f(X)\in V[X]$ is given by ${\sum }_{i\le n}{a}_i{X}^i$ then $\mathrm{res}(f)\in R[X]$ is given by ${\sum }_{i\le n}\mathrm{res}(ai)X^i$.

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

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

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