  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.
