real closed fields are o-miminal

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).

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 o-minimal.