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)?
$\aleph_1$-categorical fields are algebraically closed.
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reference-request
model-theory
1 Answers
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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.
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0@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. – 2012-04-26
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0Thanks. I have a copy of Marker, so the last bit shouldn't be a problem. – 2012-04-26