---
tag: 778ZG6P6E6
kind: definition
name: Model theory of algebraically closed fields
msc: [12L12, 03C60, 03C10]
defines: MT8TDVPBXB
uses: [0PF6TM8TWT, 04C5ZBG1WT, 1ZB13N78XW, CAEHMSX9RT]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraically_closed_field
---

## Definition

Let $\mathcal{L} = \set{+, -, \cdot, 0, 1}$ be the language of rings.
The *theory of algebraically closed fields* $\mathrm{ACF}$ is the set of first-order $\mathcal{L}$-sentences that consists of:

1. the axioms of a [[field|04C5ZBG1WT]];
2. for each $n \ge 1$, the sentence $\forall a_0 \, \forall a_1 \cdots \forall a_{n-1} \, \exists x \; x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0$.

For a prime $p$, the theory $\mathrm{ACF}_p$ is $\mathrm{ACF}$ together with the sentence $1 + 1 + \cdots + 1 = 0$ with $p$ terms.
The theory $\mathrm{ACF}_0$ is $\mathrm{ACF}$ together with, for each prime $p$, the sentence $1 + 1 + \cdots + 1 \ne 0$ with $p$ terms.
The *model theory of algebraically closed fields* is the study of these [[theories|92SWXST1ZM]] and of their models.
The models of $\mathrm{ACF}_p$ are exactly the [[algebraically closed fields|0PF6TM8TWT]] of [[characteristic|1ZB13N78XW]] $p$, for $p$ prime or $0$.

## Remarks

The basic results are these:

1. $\mathrm{ACF}$ has [[quantifier elimination|7FDDXV3JK7]] in $\mathcal{L}$.
2. For $p$ prime or $0$, the theory $\mathrm{ACF}_p$ is [[complete|655K8Y7VS8]] and decidable.
3. For $p$ prime or $0$ and every uncountable cardinal $\kappa$, any two models of $\mathrm{ACF}_p$ of cardinality $\kappa$ are isomorphic, so $\mathrm{ACF}_p$ is [[categorical|1JJCGNX240]] in $\kappa$; it is not $\aleph_0$-categorical.

Statement 3 follows from [[Steinitz's theorem|CAEHMSX9RT]]: an algebraically closed field is determined up to isomorphism by its characteristic and its transcendence degree over the prime field.
See also the theory of algebraically closed fields as an example in model theory.

Example: $\C$ is a model of $\mathrm{ACF}_0$, and the algebraic closure of $\F_p$ is a model of $\mathrm{ACF}_p$.

Non-example: $\R$ satisfies clause 1 but not clause 2, since the sentence for $n = 2$ fails at $a_1 = 0$, $a_0 = 1$: there is no real $x$ with $x^2 + 1 = 0$.