Integers, Polynomials, and Rings

by Irving, Ronald S.

• ISBN13: 9780387201726
• ISBN10: 0387201726
• Format: Paperback
• Copyright: 2003-12-01
• Publisher: Springer Verlag

Summary

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 Washington's Distinguished Teaching Award for the course on which this book is based.

Table of Contents

Preface	vii
Introduction: The McNugget Problem	1	(8)
Part I Integers
Induction and the Division Theorem	9	(14)
The Method of Induction	9	(6)
The Tower of Hanoi	15	(2)
The Division Theorem	17	(6)
The Euclidean Algorithm	23	(18)
Greatest Common Divisors	23	(4)
The Euclidean Algorithm	27	(4)
Bezout's Theorem	31	(3)
An Application of Bezout's Theorem	34	(2)
Diophantine Equations	36	(5)
Congruences	41	(16)
Congruences	41	(5)
Solving Congruences	46	(4)
Congruence Classes and McNuggets	50	(7)
Prime Numbers	57	(12)
Prime Numbers and Generalized Induction	57	(4)
Uniqueness of Prime Factorizations	61	(2)
Greatest Common Divisors Revisited	63	(6)
Rings	69	(26)
Numbers	69	(8)
Number Rings	77	(6)
Fruit Rings	83	(5)
Modular Arithmetic Rings	88	(3)
Congruence Rings	91	(4)
Euler's Theorem	95	(20)
Units	95	(4)
Roots of Unity	99	(2)
The Theorems of Fermat and Euler	101	(4)
The Euler Ψ-Function	105	(5)
RSA Encryption	110	(5)
Binomial Coefficients	115	(12)
Pascal's Triangle	115	(5)
The Binomial Theorem	120	(7)
Part II Polynomials
Polynomials and Roots	127	(14)
Polynomial Equations	127	(1)
Rings of Polynomials	128	(2)
Factoring a Polynomial	130	(3)
The Roots of a Polynomial	133	(3)
Minimal Polynomials	136	(5)
Polynomials with Real Coefficients	141	(36)
Quadratic Polynomials	141	(5)
Cubic Polynomials	146	(7)
The Discriminant of a Cubic Polynomial	153	(6)
Quartic Polynomials	159	(5)
A Closer Look at Quartic Polynomials	164	(3)
The Discriminant of a Quartic Polynomial	167	(4)
The Fundamental Theorem of Algebra	171	(6)
Polynomials with Rational Coefficients	177	(16)
Polynomials over Q	177	(4)
Gauss's Lemma	181	(3)
Eisenstein's Criterion	184	(3)
Polynomials with Coefficients in Fp	187	(6)
Polynomial Rings	193	(8)
Unique Factorization for Integers Revisited	193	(3)
The Euclidean Algorithm	196	(2)
Bezout's Theorem	198	(1)
Unique Factorization for Polynomials	199	(2)
Quadratic Polynomials	201	(20)
Square Roots	201	(3)
The Quadratic Formula	204	(5)
Square Roots in Finite Fields	209	(5)
Quadratic Field Constructions	214	(7)
Polynomial Congruence Rings	221	(20)
A Construction of New Rings	221	(5)
Polynomial Congruences	226	(4)
Polynomial Congruence Rings	230	(3)
Equations and Congruences with Polynomial Unknowns	233	(3)
Polynomial Congruence Fields	236	(5)
Part III All Together Now
Euclidean Rings	241	(14)
Factoring Elements in Rings	241	(4)
Euclidean Rings	245	(4)
Unique Factorization	249	(6)
The Ring of Gaussian Integers	255	(12)
The Irreducible Gaussian Integers	255	(4)
Gaussian Congruence Rings	259	(3)
Fermat's Theorem	262	(5)
Finite Fields	267	(14)
Primitive Roots	267	(4)
Quadratic Reciprocity	271	(6)
Classification	277	(4)
Index	281