real closed fields are o-miminal

It is clear that the axioms for a structureMathworldPlanetmath to be an ordered field can be written in L, the first order language of ordered rings. It is also true that the condition

for each odd degree polynomialPlanetmathPlanetmath pK[x], p has a root

can be written in a schema of first order sentencesMathworldPlanetmath in this languagePlanetmathPlanetmath.

Let A be all these sentences together with one that states that all positive elementsPlanetmathPlanetmath have a square root. Then one can show that the consequences of A are a complete theory T. It is clear that this theory is the theory of the real numbers. We call any L structure a real closed field (which can be defined purely algebraically also, see here (

The semi algebraic setsMathworldPlanetmath on a real closed field are Boolean combinationsPlanetmathPlanetmath of solution sets of polynomial equalities and inequalities. Tarski showed that T has quantifier eliminationMathworldPlanetmath, which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the class of semi algebraic sets being closed under projection.

Let K be a real closed field. Consider the definable subsets of K. By quantifier elimination, each is definable by a quantifier free formula, i.e. a boolean combination of atomic formulas. An atomic formula in one variable has one of the following forms:

  • f(x)>g(x) for some f,gK[x]

  • f(x)=g(x) for some f,gK[x].

The first defines a finite union of intervals, the second defines a finite union of points. Every definable subset of K is a finite union of these kinds of sets, so is a finite union of intervals and points. Thus any real closed field is o-minimal.

Title real closed fields are o-miminal
Canonical name RealClosedFieldsAreOmiminal
Date of creation 2013-03-22 13:23:04
Last modified on 2013-03-22 13:23:04
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 8
Author mathcam (2727)
Entry type Example
Classification msc 12D15
Classification msc 14P10
Classification msc 03C64
Classification msc 12D99
Related topic Theory