4 ABSTRACT ALGEBRA the text contains certain illustrative solved problems, designed to serve as templates for most of the exercises. The exercises then, in general, are designed to hone the student's skills at performing the calculations or implementing the algorithms illustrated in the template problems. Much of mathematics is facilitated by efficient algorithms and the mastery of these algorithms for arithmetic, elementary algebra, calculus, linear algebra, etc., is a foundation on which the study of higher mathematics sits. You have all achieved reasonable mastery at this level. It is time to move on. The material in this book has relatively little to do with computation and algorithm, and quite a lot to do with concept and theory-building. For this task, template problems are not helpful. What is essential is a careful and critical reading of the text and a precise assimilation of the definitions and concepts. By and large, the exercises in each section are designed to enable you to build your own understanding of the concepts incrementally. Very often exercises depend heavily on previous exercises. Sometimes you will be advised to use a certain previous exercise, other times not. Usually the exercises begin with fairly easy applications of the definitions and build up gradually. Don't look for models or templates. You have the resources within yourself to understand the concepts and do the exercises. The material in the first few chapters deals with the elementary Euclidean geometry of the plane and of 6-space. Draw pictures to help yourself visualize what the exercise is saying. Use your common sense. Many of the early exercises are intuitively obvious. Then think about how to translate your common sense into the formal language of mathematics. At first this may be challenging. Ask questions of your instructor. Work with your classmates. The level of difficulty of the exercises is uneven. Sometimes an exercise may appear so easy that you will think there is a "trick." Almost certainly there is no trick. It is just a very easy exercise. Very difficult exercises are generally designated as "Bonus Exercises," but some of the regular exercises are also quite challenging. Abstract algebra has deep and important connections to the other branches of mathematics. Sorry to say (I'm notreally sorry), you will be required to remember some basic material that you learned in earlier courses—high school geometry, linear algebra, and even a bit of calculus. Dust off your old textbooks and let's begin! TO THE INSTRUCTOR This book is somewhat terse. In consequence I recommend a more leisurely pace than might be suggested by the number of pages. In particular, Chapters 2 and 3 are par- ticularly rich in material and ideas, combining geometry, linear algebra, functions as mappings, and groups. This would certainly be indigestible were it not all grounded in the very concrete visual world of 2-dimensional and 3-dimensional geometry. Nev- ertheless these two sections warrant a slow and careful treatment. By way of contrast, Chapters 4, 5, and 6 are mostly computational algebra and serve as pleasant "comic relief" after the rich stew of Chapter 3. (Chapter 7 is once again dense with new
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