nLab theory of algebraically closed fields

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 $p$, $\mathsf{ACF}_p$ turns out to be complete, eliminates imaginaries, is stable, and admits quantifier elimination.

Definition

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

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

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

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

Properties

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

• $\mathsf{ACF}$ is stable.

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

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

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

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

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

so that $c$ is fixed by an automorphism of $\mathbb{M}$ if and only if that automorphism permutes $R$.

Related concepts

• theory of differentially closed fields?

• quantifier elimination

• stable theory?

• Morley rank?

References

• Dave Marker, Model Theory: An Introduction.