Números inteiros de Eisenstein

Abstract

Based on the development of the Theory of Integer Numbers, the present work will study of the properties, theorems, lemmas and corollaries of this theory to a more general domain, known as the Eisesntein Integer Ring, represented by Z[ω], based on the relationship between them and the ring of the Gaussian Integer,Z[i], seeking to understand in a most signi cant, simplistic and systematic way the arithmetic of this ring, constructing the notions of divisibility between two integers of Eisenstein, how to determine a common maximum divisor, how to identify the irreducible ones, and what criteria to use, why certain prime elements in Z are not irreducible in Z[ω]. We will also construct the irreducible decomposition of this ring as well as demonstrate the uniqueness of this factorization. Our interest is helping to improve a better understanding of various problems involving whole numbers and The theory of Eisenstein's Integers.