# nLab model theory of fields

### Idea

Model theory of fields is a branch of model theory dedicated to study of (axiomatic, mainly first order) theories (in the sense of logic) which are dedicated to fields and, in more recently, to differential fields.

### Basic theories of fields

First of all there is the theory of fields, which is a first order theory with equality in the language of rings $\{+, -, .,0,1\}$ and whose axioms are axioms of commutative integral domains together with an axiom

$(\forall x)(x\neq 0\implies (\exists y)(x\cdot y = 1))$

Then there are more specific theories, e.g. the theory of algebraically closed fields ACF and the theory of real closed fields RCF.

ACF has quantifier elimination and is model complete.

### Literature

• David Marker, Model theory, an introduction, Springer Grad. Texts in Math. 217 (2002)
• Model theory of fields, edited by Marker, Messmer, Pillay. Lecture Notes in Logic 5. ASL-Peters, 2006
• James Ax, The elementary theory of finite fields, Annals of Math. Series 2. 88 (2): 239–271 (1968) doi

Created on January 28, 2021 at 16:11:03. See the history of this page for a list of all contributions to it.