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)?
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This is called Macintyre's Theorem. In fact, the following are equivalent for infinite fields:
The original paper is:
Macintyre, Angus, On $\omega_1$-categorical theories of fields. Fund. Math. 71 (1971), no. 1, 1–25.