请输入您要查询的字词:

 

单词 RealClosedFieldsAreOmiminal
释义

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 (http://planetmath.org/RealClosed)).

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.

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 8:06:44