place of field

Let F be a field and an element not belonging to F.  The mapping


where k is a field, is called a place of the field k, if it satisfies the following conditions.

It is easy to see that the subring 𝔬 of the field k is a valuation domain; so any place of a field determines a unique valuation domain in the field.  Conversely, every valuation domain 𝔬 with field of fractionsMathworldPlanetmath k determines a place of k:


Let 𝔬 be a valuation domain with field of fractions k and 𝔭 the maximal idealMathworldPlanetmath of 𝔬, consisting of the non-units of 𝔬.  Then the mapping


defined by


is a place of the field k.

Proof.  Apparently,  φ-1(𝔬/𝔭)=𝔬  and the restriction  φ|𝔬  is the canonical homomorphism from the ring 𝔬 onto the residue-class ring 𝔬/𝔭.  Moreover, if  φ(x)=,  then x does not belong to the valuation domain 𝔬 and thus the inverse element x-1 must belong to it without being its unit.  Hence x-1 belongs to the ideal 𝔭 which is the kernel of the homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathφ|𝔬.  So we see that  φ(x-1)=0.


  • 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
Title place of field
Canonical name PlaceOfField
Date of creation 2013-03-22 14:56:51
Last modified on 2013-03-22 14:56:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Theorem
Classification msc 13F30
Classification msc 13A18
Classification msc 12E99
Synonym place
Synonym spot of field
Related topic KrullValuation
Related topic ValuationDeterminedByValuationDomain
Related topic IntegrityCharacterizedByPlaces
Related topic RamificationOfArchimedeanPlaces
Defines place of field