Algebraically closed $\sigma$-fields

TL;DR

This paper introduces algebraic closedness concepts for $\sigma$-fields using skew roots of skew polynomials, showing that any $\sigma$-field can embed into various algebraically closed $\sigma$-fields.

Contribution

It develops new notions of algebraic closedness for $\sigma$-fields based on skew roots, expanding the understanding of their structure and embeddings.

Findings

Every $\sigma$-field can be embedded into an algebraically closed $\sigma$-field.

Different types of algebraically closed $\sigma$-fields exist for a given $\sigma$-field.

The paper introduces a new approach using skew roots of skew polynomials.

Abstract

The concept of a skew root of a skew polynomial is used to introduce notions of algebraic closedness for $\sigma$-fields, that is, a field equipped with an endomorphism. It is shown that every $\sigma$-field can be embedded in algebraically closed $\sigma$-fields of different types.