De Morgan's law and the theory of fields

TL;DR

This paper investigates the logical properties of the classifying topos for the theory of fields, revealing it does not satisfy De Morgan's law and identifying a significant subtopos related to algebraic fields of nonzero characteristic.

Contribution

It demonstrates that the classifying topos for fields violates De Morgan's law and characterizes its largest dense De Morgan subtopos as classifying certain algebraic fields.

Findings

Classifying topos for fields does not satisfy De Morgan's law.

Largest dense De Morgan subtopos corresponds to algebraic fields of nonzero characteristic.

Provides a logical and categorical analysis of the theory of fields.

Abstract

We show that the classifying topos for the theory of fields does not satisfy De Morgan's law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.