Introductory Modern Algebra

SAUL STAHL, PhD, is Professor in the Department of Mathematics at the University of Kansas. In addition to authoring six previous books and more than thirty papers in the field of geometry, Dr. Stahl has twice been the recipient of the Carl B. Allendoerfer Award from the Mathematical Association of America.

“An in-depth explanation of the principles and practices of modern algebra in terms of the historical development from the Renaissance solution of the cubic equation to Dedekind's ideals.”  (Expofairs.com, 12 November 2015)“This book is an excellent book for an upper-level, undergraduate, one or two semester course, in modern algebra, for a typical University student population that is not especially strong in proofs.”  (MAA Reviews, 13 January 2014)

• Preface ix1 The Early History 11.1 The Breakthrough 12 Complex Numbers 92.1 Rational Functions of Complex Numbers 92.2 Complex Roots 172.3 Solvability by Radicals I 232.4 Ruler and Compass Constructibility 262.5 Orders of Roots of Unity 362.6 The Existence of Complex Numbers* 383 Solutions of Equations 453.1 The Cubic Formula 453.2 Solvability by Radicals II 493.3 Other Types of Solutions* 504 Modular Arithmetic 574.1 Modular Addition, Subtraction, and Multiplication 574.2 The Euclidean Algorithm and Modular Inverses 624.3 Radicals in Modular Arithmetic* 694.4 The Fundamental Theorem of Arithmetic* 705 The Binomial Theorem and Modular Powers 755.1 The Binomial Theorem 755.2 Fermat's Theorem and Modular Exponents 855.3 The Multinomial Theorem* 905.4 The Euler φ-Function* 926 Polynomials Over a Field 996.1 Fields and Their Polynomials 996.2 The Factorization of Polynomials 1076.3 The Euclidean Algorithm for Polynomials 1136.4 Elementary Symmetric Polynomials* 1196.5 Lagrange's Solution of the Quartic Equation* 1257 Galois Fields 1317.1 Galois's Construction of His Fields 1317.2 The Galois Polynomial 1397.3 The Primitive Element Theorem 1447.4 On the Variety of Galois Fields* 1478 Permutations 1558.1 Permuting the Variables of a Function I 1558.2 Permutations 1588.3 Permuting the Variables of a Function II 1668.4 The Parity of a Permutation 1699 Groups 1839.1 Permutation Groups 1839.2 Abstract Groups 1929.3 Isomorphisms of Groups and Orders of Elements 1999.4 Subgroups and Their Orders 2069.5 Cyclic Groups and Subgroups 2159.6 Cayley's Theorem 21810 Quotient Groups and their Uses 22510.1 Quotient Groups 22510.2 Group Homomorphisms 23410.3 The Rigorous Construction of Fields 24010.4 Galois Groups and Resolvability of Equations 25311 Topics in Elementary Group Theory 26111.1 The Direct Product of Groups 26111.2 More Classifications 26512 Number Theory 27312.1 Pythagorean triples 27312.2 Sums of two squares 27812.3 Quadratic Reciprocity 28512.4 The Gaussian Integers 29312.5 Eulerian integers and others 30412.6 What is the essence of primality? 31013 The Arithmetic of Ideals 31713.1 Preliminaries 31713.2 Integers of a Quadratic Field 31913.3 Ideals 32213.4 Cancelation of Ideals 33713.5 Norms of Ideals 34113.6 Prime Ideals and Unique Factorization 34313.7 Constructing Prime Ideals 34714 Abstract Rings 35514.1 Rings 35514.2 Ideals 35814.3 Domains 36114.4 Quotients of Rings 367A Excerpts: Al-Khwarizmi 377B Excerpts: Cardano 383C Excerpts: Abel 389D Excerpts: Galois 395E Excerpts: Cayley 401F Mathematical Induction 405G Logic, Predicates, Sets and Functions 413G.1 Truth Tables 413G.2 Modeling Implication 415G.3 Predicates and their Negation 418G.4 Two Applications 419G.5 Sets 421G.6 Functions 422Biographies 427Bibliography 431Solutions to Selected Exercises 433Index 440Notation 444