TITLE: $\aleph_1$-categorical fields are algebraically closed.
QUESTION [8 upvotes]: 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)?

REPLY [8 votes]: This is called Macintyre's Theorem. In fact, the following are equivalent for infinite fields:

$K$ is algebraically closed
$\text{Th}(K)$ is $\aleph_1$-categorical
$K$ is totally transcendental
$\text{Th}(K)$ 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.