Structures of finite fields of primes and the Euclidean square roots of unity in the finite rings

This work extends existing knowledge on the graph of square mapping a(2) equivalent to b ( mod s ) on finite fields of primes, and on finite rings, where a, b become vertices and ( a,b ) becomes an edge. Such graphs consist of cycles with binary trees attached to their vertices. The concept of arcs as k connected edges of a cycle is introduced and the product of an arc with a tree that results in a new tree is defined. This reveals another aspect of prime finite fields which is equivalent to the well-known product of the kernel and the image of a function. Next, we present the theorem of Euclidean roots of unity in a ring Z(sp), where s and p are primes. Z(sp) is analysed and partitioned into three sets. Two of them contain multiples of the two primes and the third one has interesting properties that are later described in this paper. A number of new theorems are proved in the process that lay the foundation for the novel ideas presented in this work. The generalized theorem of the Euclidean roots of unity in a finite ring is the basis of this work and it presents a way of calculating all square roots of unity and fixed points in a finite ring under the square mapping.