real closed fields RealClosedFields 2003-01-23 11:00:15 2007-07-10 03:28:37 Example o-minimality real closed field % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here It is clear that the axioms for a structure to be an ordered field can written in $L$, the first order language of ordered rings. It is also true that the following conditions can be written in a schema of first order sentences in this language. For each odd degree polynomial $p \in K[x]$, $p$ has a root. Let $A$ be all these sentences together with one that states that all positive elements 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 {\em real closed field}. \medskip The {\em semi algebraic sets} on a real closed field are Boolean combinations of solution sets of polynomial equalities and inequalities. Tarski showed that $T$ has quantifier elimination, which is equivalent to the class of semi algebraic sets being closed under projection. \medskip 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: \begin{itemize} \item $f(x)>g(x)$ for some $f,g \in K[x]$ \item $f(x)=g(x)$ for some $f,g \in K[x]$. \end{itemize} 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.