Integer and Polynomial Algebra

This book is a concrete introduction to abstract algebra and number theory. Starting from the basics, it develops the rich parallels between the integers and polynomials, covering topics such as Unique Factorization, arithmetic over quadratic number fields, the RSA encryption scheme, and finite fields. In addition to introducing students to the rigorous foundations of mathematical proofs, the authors cover several specialized topics, giving proofs of the Fundamental Theorem of Algebra, the transcendentality of (e), and Quadratic Reciprocity Law. The book is aimed at incoming undergraduate students with a strong passion for mathematics. Title Contents Preface Chapter 1. The Integers 1.1. Basic Properties 1.2. Well Ordering Principle 1.3. Primes 1.4. Many Primes 1.5. Euclidean Algorithm 1.6. Factoring Integers 1.7. Irrational Numbers 1.8. Unique Factorization in More General Rings Notes on Chapter 1 Chapter 2. Modular Arithmetic 2.1. Linear Equations 2.2. Congruences 2.3. The Ring \bZ_{n} 2.4. Equivalence Relations 2.5. Chinese Remainder Theorem 2.6. Congruence Equations 2.7. Fermat’s Little Theorem 2.8. Euler’s Theorem 2.9. More on Euler’s Phi Function 2.10. Primitive Roots Notes on Chapter 2 Chapter 3. Diophantine Equations and Quadratic Number Domains 3.1. Pythagorean Triples 3.2. Fermat’s Equation for n=4 3.3. Quadratic Number Domains 3.4. Pell’s Equation 3.5. The Gaussian Integers 3.6. Quadratic Reciprocity Notes on Chapter 3 Chapter 4. Codes and Factoring 4.1. Codes 4.2. The Rivest-Shamir-Adelman Scheme 4.3. Primality Testing 4.4. Factoring Algorithms Notes on Chapter 4 Chapter 5. Real and Complex Numbers 5.1. Real Numbers 5.2. Complex Numbers 5.3. Polar Form 5.4. The Exponential Function 5.5. Fundamental Theorem of Algebra 5.6. Real Polynomials Notes on Chapter 5 Chapter 6. The Ring of Polynomials 6.1. Preliminaries on Polynomials 6.2. Unique Factorization for Polynomials 6.3. Irreducible Polynomials in \bZ[x] 6.4. Eisenstein’s Criterion 6.5. Factoring Modulo Primes 6.6. Algebraic Numbers 6.7. Transcendental Numbers 6.8. Sturm’s Algorithm 6.9. Symmetric Functions 6.10. Cubic Polynomials Notes on Chapter 6 Chapter 7. Finite Fields 7.1. Arithmetic Modulo a Polynomial 7.2. An Eight-Element Field 7.3. Fermat’s Little Theorem for Finite Fields 7.4. Characteristic 7.5. Algebraic Elements 7.6. Finite Fields 7.7. Automorphisms of \bF_{p^{d}} 7.8. Irreducible polynomials of all degrees 7.9. Factoring Algorithms for Polynomials 7.10. Factoring Rational Polynomials Notes on Chapter 7 Bibliography Index