model theory of fields

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.

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.

- 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

category: model theory, algebra

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