INTRODUCTION 3 The traditional approach has the advantage of efficiency but the defect of severing the material from its historical roots and from its connections with the rest of the body of mathematics. This text attempts a different approach, letting the abstract concepts emerge gradually from less abstract problems about geometry, polynomials, numbers, etc. This is how the subject evolved historically. This is how all good mathematics evolves—abstraction and generalization is forced on us as we attempt to understand the "concrete" and the particular. In many ways this book is a throwback to a less abstract algebra, driven by the problems that fascinated Lagrange, Gauss, and Abel. They helped fashion a tool—the group—to solve their problem. Galois began the paradigm shift in which the study of the internal structure of the tool itself became more interesting than the solution of the original problem. This is the beginning of abstract algebra and the end of our text. In that sense our text should be called An Invitation to Abstract Algebra, and the reader who finds the subject enchanting is advised to study further in texts such as Topics in Algebra by Herstein. Section I begins with Euclid's geometry and anachronistically teases out the con- cept of a group from his intuitive treatment of congruence. Here symmetry appears in its most visual form with the regular polygons and polyhedra. Later the "hidden symmetries" of the regular polygons will appear in the work of Gauss. Section II begins the main theme of polynomial equations much as they were stud- ied in Western Europe from the 1500s through the 1700s, and we watch the true historical emergence of the concept of a group as it developed in the conversation of mathemati- cians in the late 18th and early 19th centuries. It is appropriate here to acknowledge a debt and to recommend enthusiastically the book Galois' Theory of Algebraic Equa- tions by Jean-Pierre Tignol, which does a superb and detailed job of tracing how this mathematics was made. In Section III we turn to the elementary theory of numbers, beginning with Euclid but focusing on its modern reemergence in the cryptic correspondence of Pierre de Fermat and its clarification in the work of Leonhard Euler and Karl Gauss. The concepts of domain and ring emerge as organizing principles and help clarify the similarities and differences between numbers and polynomials. From 1644 to 1994, from Fermat to Andrew Wiles, the Dulcinea of every number theorist's quest was the "Last Theorem" of Fermat. Much of the work of Wiles and its immediate antecedents lies in the domain of still-modern algebra, where this text dares not tread. Nevertheless we do present some older work that helped clarify the important concept of unique factorization. Finally we return to the subject of polynomial equations and study some of the astonishing work of Karl Gauss on cyclotomic equations and the work of Evariste Galois, which has come to be known as Galois Theory. Here the concept of a field comes to the fore and a remarkable correspondence between fields and groups emerges. ADVICE TO STUDENTS A few words are in order about this book as a learning tool. It is likely that most if not all of your previous math textbooks have employed the following format. Each section of
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