Numbers & Ideals. An Introduction to Some Basic Concepts of Algebra and Number Theory 12+

📓 This is a basic book of mathematics covering all of mathematics up to but not including calculus. It is suitable for freshman college students or advanced high school students.This book bridges the gap between abstract mathematics and and concrete application in Modern Algebra and Modern Number Theory. It provides an introduction to some basic notions of Algebra and Number Theory, such as rings, fields and ideals, grouped around the theory of algebraic integers in quadratic number theories.Starting at a thoroughly elementary level, the book discusses the basic properties of the natural numbers, including primes, prime factorization and the notion of convergence. This discussion is followed by the introduction of the notion of integers in quadratic number fields and it is shown that the law of unique prime factorization may not hold in such fields. Ideals and then introduced as a means of reestablishing the law of unique factorization in a different setting.Numbers and Ideals is intended as a text for a new kind of introductory course, given, in the freshman or sophomore year, as a preliminary to advanced courses given in the freshman or sophomore year, as a preliminary to advanced courses in Algebra and Number Theory . It is also excellent for selected classes at the senior high school level and as self-study material for teachers.