On Cohesive Products of Fields

TL;DR

This paper develops the theory of cohesive products of fields using computability theory, exploring their Galois groups, first-order theories, and automorphisms, extending ultraproduct concepts to a computability context.

Contribution

It introduces cohesive products as computability-theoretic analogs of ultraproducts for fields, and analyzes their Galois groups, theories, and automorphisms.

Findings

Characterized infinite Galois groups of cohesive powers

Described hyper-automorphism groups of cohesive powers

Analyzed first-order theories of cohesive powers of number fields

Abstract

We develop the foundations of effective ultraproducts of fields and their Galois groups using the methods of computability theory. These computability-theoretic analogs of ultraproducts are called cohesive products, since the role of an ultrafilter is played by a cohesive set. A set of natural numbers is cohesive if it is infinite and cannot be partitioned into two infinite subsets by any computably enumerable set. In particular, we investigate the way cohesive products interact with field extensions with emphasis on both finite and infinite Galois extensions, and the associated Galois groups. We study the first-order theories and definability of cohesive powers of number fields, and characterize the infinite Galois groups of cohesive powers for a large class of infinite Galois extensions. Finally, we introduce hyper-automorphisms, which are automorphisms of a cohesive power that respect non-standard field operations, and give a complete description of the hyper-automorphism groups of cohesive powers of a large class of computable Galois extensions, and use them to describe the classical infinite Galois groups of such fields.