  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.
