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 $mathbb{Z}diagup 2mathbb{Z}$ as a residual field. One of the import
ant 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.
