PREFAC E There are numerous excellent textbooks presenting the basics of abstract algebra for college students. Why write another? Most of the existing texts have a foundational character. They seem designed to lay a firm foundation for future graduate courses in abstract algebra. This is a noble goal and well served by many of the exemplars. There is however a large audience for an undergraduate abstract algobra course consisting of students who will likely never take a graduate course in abstract algebra. Notable among these are the future high school math- ematics teachers. These students are better served by a course that emphasizes the roots of abstract algebra, which live in the rich soil of high school mathematics—Euclidean geometry, polynomial algebra, and trigonometry. Out of this soil spring naturally the con- cepts of symmetry, the complex numbers and the cyclotomic number fields, eventually blossoming into the Galois theory of equations. The intention of this text is to emphasize the organic and historical development of the abstract theory of groups, rings, and fields from the substrate of high school mathematics. In Part I the "history" is fictitious. It is only with imaginative hindsight that we can attribute the concept of a group of motions to Euclid. In the later parts, however, the history is genuine, although the notation and terminology is updated. Novel and exciting ideas and theorems are encountered early and often—the 2-dimensional symmetry groups in Chapter 3, Cardano's formulas in Chapter 5, the complex numbers in Chapter 6, the Fundamental Theorem of Algebra in Chapter 7 and the 3-dimensional symmetry groups in Chapter 8. Section 3 on Number Theory fea- tures the work of Fermat. Not only his Little Theorem but also the Two Squares and Four Squares Theorems, as well as some cases of the celebrated Last Theorem and its polynomial analogue, are presented. The final Grand Synthesis section begins with a careful treatment of Gauss' proof of the straight-edge-and-compass constructibility of the regular 17-sided polygon, and culminates with Galois' theory of equations. Con- structibility of regular polygons is a lovely topic, wonderfully down-to-earth and visual, yet laden with deep connections to subtle topics in number theory and group theory. A course that ends with Chapter 16 (even omitting some of the earlier material) will have presented a rich array of ideas to the students, all closely tied to the most elementary of questions in Euclidean geometry and the study of numbers and polynomial equations. At the end of the Introduction, I discuss some possible syllabi for semester-long and year-long courses. Typically, Galois Theory appears as the grand finale and raison d'etre for a first course in abstract algebra. But all too often there is not enough time to reach the finale ix
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