nLab
theory of algebraically closed fields

Contents

Idea

Fields are finitely first-order axiomatizable. An algebraically closed field is further axiomatized by infinitely many sentences which state that all non-constant polynomials have a root. Once we additionally specify a characteristic pp, ACF p\mathsf{ACF}_p turns out to be complete, eliminates imaginaries, is stable, and admits quantifier elimination.

Definition

ACF\mathsf{ACF} is the countable collection of sentences in the language ring\mathcal{L}_{\operatorname{ring}} of rings given by:

{field axioms}+{(a 0,,a n1)(x)[x n+ i=0 n1a ix i i=0]},\{\text{field axioms}\} + \left\{ (\forall a_0, \ldots, a_{n-1}) (\exists x)[x^n + \sum_{i=0}^{n-1} a_i x_i^i = 0] \right\},

where n=1,2,n = 1, 2, \ldots.

We can additionally specify a characteristic pp to obtain ACF p\mathsf{ACF}_p by either adding the axioms {10,1+10,}\{ 1 \ne 0 , 1+1 \ne 0 , \cdots \} to get ACF 0\mathsf{ACF}_0 or adding the axiom 1++1pterms=0\underset{p\; terms}{\underbrace{1 + \cdots + 1}} = 0 to get ACF p\mathsf{ACF}_p (where pp is prime).

Properties

  • ACF\mathsf{ACF} has quantifier elimination. This amounts to a special case of Chevalley’s direct image theorem from algebraic geometry.

  • ACF\mathsf{ACF} is stable.

  • ACF\mathsf{ACF} is totally transcendental: Morley rank? is defined everywhere. In this setting Morley rank subsumes the usual Krull dimension of an algebraic variety.

  • ACF\mathsf{ACF} is uncountably categorical: for each uncountable κ\kappa, models of ACF\mathsf{ACF} of size κ\kappa must all be isomorphic. This fails at 0\aleph_0: alg\mathbb{Q}^{\operatorname{alg}} has transcendence degree zero while there exist countable algebraically closed overfields of \mathbb{Q} with infinite transcendence degree.

  • ACF\mathsf{ACF} eliminates imaginaries. This means that its syntactic category Def(ACF)\mathbf{Def}(\mathsf{ACF}) has effective internal congruences, and has a good Galois theory.

  • It’s easy to see that ACF\mathsf{ACF} codes all finite sets: if RR is a finite set of points inside a monster 𝕄\mathbb{M}, its code is the tuple cc of coefficients for the polynomial

f(X)=df rR(Xr),f(X) \overset{\operatorname{df}}{=} \displaystyle \prod_{r \in R} (X - r),

so that cc is fixed by an automorphism of 𝕄\mathbb{M} if and only if that automorphism permutes RR.

References

  • Dave Marker, Model Theory: An Introduction.

Last revised on February 19, 2018 at 13:40:39. See the history of this page for a list of all contributions to it.