real closed fields are omiminal
It is clear that the axioms for a structure^{} 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 polynomial^{} $p\in K[x]$, $p$ has a root
can be written in a schema of first order sentences^{} in this language^{}.
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 real closed field (which can be defined purely algebraically also, see here (http://planetmath.org/RealClosed)).
The 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.
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 $f,g\in K[x]$

•
$f(x)=g(x)$ for some $f,g\in K[x]$.
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 ominimal.
Title  real closed fields are omiminal 

Canonical name  RealClosedFieldsAreOmiminal 
Date of creation  20130322 13:23:04 
Last modified on  20130322 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 