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Math 229: Model Theory
Homework # 5

Date: 23 October 2003

1. We say that a field $K$ is orderable if there exists an ordering $<$ on $K$ making $(K,+,\times,<,0,1)$ into an ordered field. Give two proofs that the class of orderable fields is elementary. That is, give a soft proof of first-order axiomatizability using no algebra at all and then also give an explicit axiomatization (and a proof of the correctness of your axiomatization).

2. Show that a field $R$ has an expansion to the language of ordered rings in which $R$ is a real closed field if and only if $-1$ is not a square in $R$, every odd degree polynomial over $R$ has a root in $R$, and the nonzero squares have index two in the multiplicative group of $R$.

3. Show that there are only finitely many homeomorphism types of real affine Fermat curves, $C_n({\mathbb{R}}) := \{ \langle x, y \rangle \in {\mathbb{R}}^2 \vert  x^n + y^n = 1 \}$, as $n$ runs through the positive integers. [You may assume Wilkie's theorem.]

4. Let $(M,<,\ldots)$ be an o-minimal structure and $A \subseteq M$ a subset of the universe. Show that there is a unique (up to isomorphism) prime model over $A$. Is the prime model necessarily minimal?

5. Classify the $\aleph_0$-categorical o-minimal theories.