Model theory of algebraically closed fields

Builds on Algebraically closed field Field Characteristic of a field Steinitz's theorem on algebraically closed fields

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:

the axioms of a field;
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 and of their models. The models of $\mathrm{ACF}_p$ are exactly the algebraically closed fields of characteristic $p$ , for $p$ prime or $0$ .

Remarks

The basic results are these:

$\mathrm{ACF}$ has quantifier elimination in $\mathcal{L}$ .
For $p$ prime or $0$ , the theory $\mathrm{ACF}_p$ is complete and decidable.
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 in $\kappa$ ; it is not $\aleph_0$ -categorical.

Statement 3 follows from Steinitz's theorem: 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 $x^2 + 1 = 0$ .

Reading path

To understand this page from the ground up, read these 82 pages in order. The foundations come first.

Algebraically closed field concept
Characteristic of a field concept
Element definition
Field concept
Group concept
Proposition concept
Transcendence basis concept
Transcendence degree concept
Set definition
Ring definition
Implication definition
Predicate definition
Ordered pair definition
Commutative ring definition
Polynomial ring definition
Ring homomorphism definition
Subring definition
Unit of a ring definition
Zero divisor definition
Ideal definition
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Divisibility in a commutative ring definition
Degree of a polynomial definition
Polynomial evaluation definition
Field definition
Kernel of a ring homomorphism definition
Ring isomorphism definition
Subset definition
Integral domain definition
Ideal generated by a subset definition
Maximal ideal definition
Two-sided ideal definition
Uniqueness quantifier definition
Leading coefficient definition
Evaluation homomorphism definition
Root of a polynomial definition
Binary relation definition
Subfield definition
The only ideals of a field are zero and the whole field statement
Quotient ring definition
Irreducible element definition
Euclidean domain definition
Principal ideal definition
Monic polynomial definition
Split polynomial definition
Degree of a product statement
Field extension definition
Irreducible polynomial definition
Universal property of the quotient ring statement
Correspondence theorem for rings statement
Function definition
Principal ideal domain definition
Division algorithm for polynomials statement
Algebraic element definition
Algebraically closed field definition
Intermediate field definition
R[x] is an integral domain iff R is statement
First isomorphism theorem for rings statement
R/M is a field iff M is maximal statement
Codomain definition
Injective function definition
In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
Minimal polynomial of an algebraic element definition
Algebraic extension definition
Field generated by adjoining elements definition
K[x] is a Euclidean domain statement
Surjective function definition
The minimal polynomial divides every polynomial vanishing at the element statement
Algebraic closure definition
Simple extension definition
Splitting field definition
K[x]/(f) is a field iff f is irreducible statement
Bijective function definition
K(a) is isomorphic to K[x] modulo the minimal polynomial statement
Equinumerous sets definition
Cardinality definition
Uniqueness of splitting fields statement
Uniqueness of the algebraic closure statement
Steinitz's theorem on algebraically closed fields statement
Model theory of algebraically closed fields definition

Tag	`778ZG6P6E6`
URI	`m2://778ZG6P6E6`
Kind	definition
Writing	W0 stub
MSC	12L12, 03C60, 03C10
Defines	Model theory of algebraically closed fields
Uses	Algebraically closed field, Field, Characteristic of a field, Steinitz's theorem on algebraically closed fields
Changed	2 November 2026

Model theory of algebraically closed fields

Definition

Remarks

Reading path

Further reading

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