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Abstract
A partition over a finite field is defined and each equivalence class (fundamental class) is constructed and represented by a set called a fundamental set. If a primitive element is used to construct the addition table over one fundamental set of each fundamental class, then all additions over the field can be computed. The number of fundamental classes is given for some finite fields. The solutions of P(x) = x('p('n)) + ax + b, P(x) (ELEM) Z(,p){x} in the field are discussed.
This partition is extended to define a new equivalence relation over fundamental classes to minimize the computational time and the number of fundamental sets required to carry out the additions over the field. Some results on the solutions of the trinomial h(x) = x('p('(lamda))) + ax + b (ELEM) Z{x} are discussed.
The additive subgroups of a field are discussed and a construction method is developed by using the fundamental sets. The number of additive subgroups is given under certain conditions.
