Ternary number systems in finite fields

The work continues the author's previous study of positional number systems in finite fields. The paper considers ternary number systems and arithmetic operations algorithms for the representation of elements of finite fields in the so-called ternary reduced number systems, which are reductions of the canonical number systems when mapping the corresponding ring of integers of a quadratic field into some prime field. A classification of finite fields in which such number systems exist is given. It is proved that the reduced ternary number systems exist for most finite prime fields.