# $\aleph_1$-categorical fields are algebraically closed.

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:

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.

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@t.b. Thanks for adding the links. Brett: I should maybe point out that the theorem proven in these references is that totally transcendental implies algebraically closed. The remaining step that $\aleph_1$-categorical implies totally transcendental is part of the standard presentation of Morley's Theorem, which can be found in Marker or Hodges for example. – Alex Kruckman Apr 26 '12 at 5:09
Thanks. I have a copy of Marker, so the last bit shouldn't be a problem. – Brett Frankel Apr 26 '12 at 16:23