Extensions of special 3-fields

TL;DR

This paper explores the structure and automorphisms of finite extensions of unital 3-fields, revealing significant differences from classical field theory and introducing new methods to construct such fields.

Contribution

It provides a detailed analysis of finite extensions of unital 3-fields, clarifies their automorphism groups, and introduces novel construction techniques using product structures.

Findings

Finite extensions of unital 3-fields differ markedly from classical fields.

Automorphism groups of these extensions are characterized.

New unital 3-fields can be constructed via product structures, including Cartesian products.

Abstract

We investigate finite field extensions of the unital 3-field, consisting of the unit element alone, and find considerable differences to classical field theory. Furthermore, the structure of their automorphism groups is clarified and the respective subfields are determined. In an attempt to better understand the structure of 3-fields that show up here we look at ways in which new unital 3-fields can be obtained from known ones in terms of product structures, one of them the Cartesian product which has no analogue for binary fields.