Computational Problems And Properties Over The Rings Of Some Algebraic Integers

In this paper, we study the structures and properties of the fields of some quadratic and cubic algebraic numbers and the ones of the rings of their algebraic integers. This pa-per is divided into two parts. In the first part, we prove that the ring Z[u]={a+bu}a, b∈Z} is a Euclidean domain by the theory of optimization, and provide the characterization of inverse elements and prime elements in Z[u] and the methods of the prime factorization of elements which are not zero or unit in Z[u], where u=1/2+(?)i∈C. Furthermore, we study the residue class ring of Z[u] and and its representatives, and show the neces-sary and sufficient condition on which the residue class ring is an involutive ring. In the second part, we mainly study several special cubic algebraic number fields, and provide their integral basis and the expression of the fields. Furthermore, we prove that the al-gebraic integer ring is isomorphic to a matix ring, and present their property of unique factorization and an algorithm for all different representatives of elements in their quotient ring.