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

The map $\mathrm{res}:V\to V/\mu$ 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 ${\sum }_{i\le n}{a}_i{X}^i$ 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 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