Structure of unital 3-fields

TL;DR

This paper explores the algebraic structure of unital 3-fields, showing they are isomorphic to invertible elements in certain local rings with specific residual fields, and clarifies their differences from traditional binary fields.

Contribution

It provides a novel characterization of unital 3-fields as invertible elements in local rings with residual field Z/2Z, including examples and structural insights.

Findings

3-fields are isomorphic to invertible elements in local rings

Finite 3-fields are structurally distinct from binary fields

Intrinsic characterization of the maximal ideal in the associated local ring

Abstract

We investigate fields in which addition requires three summands. These ternary fields are shown to be isomorphic to the set of invertible elements in a local ring $\mathcal{R}$ having $\mathbb{Z}\diagup 2\mathbb{Z}$ as a residual field. One of the important technical ingredients is to intrinsically characterize the maximal ideal of $\mathcal{R}$. We include a number of illustrative examples and prove that the structure of a finite 3-field is not connected to any binary field.