In this chapter we introduce the algebraic numbers as solutions of polynomial equations with integer coefficients. Among these numbers, the major players are the solutions of equations with integer coefficients whose leading coefficient is 1. These are the algebraic integers. We develop a theory of factorization of algebraic integers, analogous to factorization of whole numbers. In many ways the theories are alike, but in at least one essential way-uniqueness of factorization-there are differences.