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เนื้อหาย่อ : PREFACE This book is intended for a one-year undergraduate course in abstract algebra. Its design is such that the book can also be used for a one-semester course. The book contains more material than normally would be taught in a one-year course. This should give the teacher flexibility with respect to the selection of the content and the level at which the book is to be used. We give a rigorous treatment of the fundamentals of abstract algebra with numerous examples to illustrate the concepts. It usually takes students some time to become comfortable with the seeming abstractness of modern algebra. Hence we begin at a leisurely pace paying great attention to the clarity of our proofs. The only real prerequisite for the course is the appropriate mathematical maturity of the students. Although the material found in calculus is independent of that of abstract algebra, a year of calculus is typically given as a prerequisite. Since many of the examples in algebra comes from matrices, we assume that the reader has some basic knowledge of matrix theory. The book should prepare the student for higher level mathematics courses and computer science courses. We have many problems of varying difficulty appearing after each section. We occasionally leave as an exercise the verifcation of a certain point in a proof. However, we do not rely on exercises to introduce concepts which will be needed later on in the text. Topics are introduced that have never appeared in this type of textbook. They include Grobner basis, rings of matrices, and Noetherian and Artinian rings. Another distinguishing feature of the book is the Worked-Out Exercises which appear after every section. These Worked-Out Exercises provide not only techniques of problem solving, but also supply additional information to enhance the level of knowledge of the reader. For example, in Chapter 7, we illustrate several techniques that are very effective in determining the Sylow subgroups of a group, whether the group is simple or not, and in determining the structure of a group. In Chapter 9, we give numerous examples and show how to determine different Abelian groups of a given order. We also show how to fnd the elementary divisors, the torsion coefficients, and the betti number of a fnitely generated Abelian group. In Chapter 15, we give an algorithmic procedure to find the greatest common divisor and illustrate it in full detail.