Abstract Algebra with Applications

Description

Abstract Algebra with Applications provides a friendly and concise introduction to algebra, with an emphasis on its uses in the modern world. The first part of this book covers groups, after some preliminaries on sets, functions, relations, and induction, and features applications such as public-key cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry and physics. The second part of this book covers rings and fields, and features applications such as random number generators, error correcting codes, the Google page rank algorithm, communication networks, and elliptic curve cryptography. The book's masterful use of colorful figures and images helps illustrate the applications and concepts in the text. Real-world examples and exercises will help students contextualize the information. Intended for a year-long undergraduate course in algebra for mathematics, engineering, and computer science majors, the only prerequisites are calculus and a bit of courage when asked to do a short proof. Contents List of Figures Preface PART I GROUPS 1 Preliminaries 1.1 Introduction 1.2 Sets 1.3 The Integers 1.4 Mathematical Induction 1.5 Divisibility, Greatest Common Divisor, Primes, and Unique Factorization 1.6 Modular Arithmetic, Congruences 1.7 Relations 1.8 Functions, the Pigeonhole Principle, and Binary Operations 2 Groups: A Beginning 2.1 What is a Group? 2.2 Visualizing Groups 2.3 More Examples of Groups and Some Basic Facts 2.4 Subgroups 2.5 Cyclic Groups are Our Friends 3 Groups: There’s More 3.1 Groups of Permutations 3.2 Isomorphisms and Cayley’s Theorem 3.3 Cosets, Lagrange’s Theorem, and Normal Subgroups 3.4 Building New Groups from Old, I: Quotient or Factor Groups G/H 3.5 Group Homomorphism 3.6 Building New Groups from Old, II: Direct Product of Groups 3.7 Group Actions 4 Applications and More Examples of Groups 4.1 Public-Key Cryptography 4.2 Chemistry and the Finite Fourier Transform 4.3 Groups and Conservation Laws in Physics 4.4 Puzzles 4.5 Small Groups PART II RINGS 5 Rings: A Beginning 5.1 Introduction 5.2 What is a Ring? 5.3 Integral Domains and Fields are Nicer Rings 5.4 Building New Rings from Old: Quotients and Direct Sums of Rings 5.5 Polynomial Rings 5.6 Quotients of Polynomial Rings 6 Rings: There’s More 6.1 Ring Homomorphisms 6.2 The Chinese Remainder Theorem 6.3 More Stories about F[x] Including Comparisons with Z 6.4 Field of Fractions or Quotients 7 Vector Spaces and Finite Fields 7.1 Matrices and Vector Spaces over Arbitrary Fields and Rings like Z 7.2 Linear Functions or Mappings 7.3 Determinants 7.4 Extension Fields: Algebraic versus Transcendental 7.5 Subfields and Field Extensions of Finite Fields 7.6 Galois Theory for Finite Fields 8 Applications of Rings 8.1 Random Number Generators 8.2 Error-Correcting Codes 8.3 Finite Upper Half Planes and Ramanujan Graphs 8.4 Eigenvalues, Random Walks on Graphs, and Google 8.5 Elliptic Curve Cryptography References Index