Integers Polynomials and Rings : A Course in Algebra Mathematics is often regarded as the study of calculation but in fact mathematics is much more. It combines creativity and logic in order to arrive at abstract truths. This book is intended to illustrate how calculation creativity and logic can be combined to solve a range of problems in algebra. Originally conceived as a text for a course for future secondary school mathematics teachers this book has developed into one that could serve well in an undergraduate course in abstract algebra or a course designed as an introduction to higher mathematics. Not all topics in a traditional algebra course are covered. Rather the author focuses on integers polynomials their ring structure and fields with the aim that students master a small number of serious mathematical ideas. The topics studied should be of interest to all mathematics students and are especially appropriate for future teachers. One nonstandard feature of the book is the small number of theorems for which full proofs are given. Many proofs are left as exercises and for almost every such exercise a detailed hint or outline of the proof is provided. These exercises form the heart of the text. Unwinding the meaning of the hint or outline can be a significant challenge and the unwinding process serves as the catalyst for learning. Ron Irving is the Divisional Dean of Natural Sciences at the University of Washington. Prior to assuming this position he served as Chair of the Department of Mathematics. He has published research articles in several areas of algebra including ring theory and the representation theory of Lie groups and Lie algebras. In 2001 he received the University of Washingtons Distinguished Teaching Award for the course on which this book is based. Introduction: The McNugget Problem Introduction: The McNugget Problem Integers Induction and the Division Theorem The Euclidean Algorithm Congruences Prime Numbers Rings Euler’ Theorem Binomial Coefficients Polynomials Polynomials and Roots Polynomials with Real Coefficients Polynomials with Rational Coefficients Polynomial Rings Quadratic Polynomials Polynomial Congruence Rings All Together Now Euclidean Rings The Ring of Gaussian Integers Finite Fields.