The Fundamental Theorem Of Arithmetic

Abstract

Abstract Algebraic numbers and integers. In this chapter we consider some simple generalizations of the notion of an integer. We defined an algebraic number in § 11.5; ξis an algebraic number if it is a root of an equation then ξ is said to be an algebraic integer. This is the natural definition, since a rational ξ a/b satisfies bξ a 0, and is an integer when b 1. When n2, ξis said to be a quadraticnumber, or integer, as the case may be. These definitions enable us to restate Theorem 45 in the form Theorem 206. An algebraic integer, if rational, is a rational integer. † We defined the ‘rational integers’ in § 1.1. Since then we have described them simply as the ‘integers’, but now it becomes important to distinguish them explicitly from integers of other kinds.