$\aleph_1$-categorical fields are algebraically closed.

Question

I'd like to understand the proof that if $K$ is an infinite field the theory of $K$ is $\aleph_1$-categorical, then $K$ is algebraically closed--but I'm having trouble finding it in the literature. Any ideas where I can find the appropriate paper(s)?

Answer 1

This is called Macintyre's Theorem. In fact, the following are equivalent for infinite fields:

1. $K$ is algebraically closed
2. $\text{Th}(K)$ is $\aleph_1$-categorical
3. $K$ is totally transcendental
4. $\text{Th}(K)$ has quantifier elimination

The original paper is:

Macintyre, Angus, On $\omega_1$-categorical theories of fields. Fund. Math. 71 (1971), no. 1, 1–25.

It also appears as Theorem 3.1 in Stable Groups by Poizat.