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