INTRODUCTION 5 ideas.) I believe that Chapter 6 deserves careful attention. To my amazement students usually enter this upper-level course with essentially no knowledge of the complex numbers. At Ohio State, we have two parallel year-long abstract algebra sequences, one designated "honors", the other not. This text is used in the non-honors version and I think of the typical student as a future high school math teacher. For this reason, I have tried to link the material to important themes in high school mathematics—Euclidean geometry, polynomials, numbers, and functions. In a year-long course, I still do not in general cover all of the material in the book. Indeed, I am satisfied if I get through Chapter 16 and delighted if I get through Chapter 18, while omitting the optional Chapters 9A and 15A. If you are teaching a one-semester course, I think it is realistic (though I have never tried this) to cover the following material: Chapters 1-8, 10-12, and 16. Chapter 7 can be given a "light", heuristic treatment. It may be viewed more as a historical interlude rather than a body of material that needs to be presented rigorously and digested fully. In any case, this set of chapters gives a nice sampling of geom- etry, polynomial algebra, number theory and group theory, with highlights including Fermat's Little Theorem, Lagrange's Theorem, the Fundamental Theorem of Algebra and Gauss' proof of the constructibility of the regular 17-gon. Rings get short shrift. If your students already know some or all of the basic number theory in Chapters 10- 12, then you might want to add Chapters 14 or 15 to touch on some ring theory. Alternatively Chapters 17 and 18 will deepen their grasp of linear algebra and field theory. It seems impossible to write an algebra text at this level without culminating in the beautiful Galois Theory, and likewise almost impossible to cover this material adequately, except with honors students. I consider it an acceptable loss if future high school math teachers do not see Galois Theory in this course, though of course I cherish the hope that they will be inspired to keep the book and read it themselves some day. Instead of the usual procedure of placing all of the exercises for a section at the end of the section, I have interwoven the exercises into the body of the text. They are however numbered sequentially throughout each section. I apologize for the extra effort this entails in finding the exercises. As mentioned in the advice to the students above, exercises very often build on previous exercises, even if this is not explicitly indicated in the text. Skipping some exercises is a peril. It may make later otherwise easy exercises difficult, if not impossible, for the students. I have learned that students need to learn that it is not necessary, indeed counterproductive, to keep reinventing the wheel. Theorems in the text and earlier exercises are meant to be used as tools to do later exercises. Usually, I do one or two of the exercises in class for the students, either before or shortly after they are assigned, by way of illustration or template, especially in the early part of the course. Then I usually give out solutions for most or all of the exercises, after the students have submitted their work. This material presents two challenges for the students. For me the most important is the intuitive assimilation of the concepts, the internalization of such abstractions as group,
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