henselian field

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

The set μ={x:|x|<1} is a maximal idealMathworldPlanetmathPlanetmath of V. The factor R:=V/μ is called the residue fieldMathworldPlanetmath or the residue class field.

The map res:VV/μ given by xx+μ 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)V[X] is given by inaiXi then res(f)R[X] is given by inres(ai)Xi.

Hensel property: Let f(x)V[x]. Suppose res(f)(x) has a simple root ek. Then f(x) has a root eV and res(e)=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 isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath) smallest henselian field Kh containing it. We call Kh 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