Structure of unital 3-fields

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.