# henselian field

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 $\operatorname{res}:V\to V/\mu$ 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 $\operatorname{res}(f)\in R[X]$ is given by $\sum_{i\leq n}\operatorname{res}(a{i})X^{i}$.

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

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 $K$.

 Title henselian field Canonical name HenselianField Date of creation 2013-03-22 13:28:37 Last modified on 2013-03-22 13:28:37 Owner mps (409) Last modified by mps (409) Numerical id 9 Author mps (409) Entry type Definition Classification msc 13F30 Classification msc 13A18 Classification msc 11R99 Classification msc 12J20 Related topic Valuation Related topic ValuationDomainIsLocal Related topic ValuationRingOfAField Defines valuation ring Defines residue field Defines residue class field Defines Hensel property Defines henselian Defines henselisation