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Library of Congress Cataloging-in-Publication Data:
Dixon, Martyn R. (Martyn Russell), 1955- author.
An introduction to essential algebraic structures / Martyn R. Dixon, Department of Mathematics, The University of Alabama, Tuscaloosa, AL, Leonid A. Kurdachenko, Department of Algebra, National University of Dnepropetrovsk, Dnepropetrovsk, Ukraine, Igor Ya. Subbotin, Department of Mathematics and Natural Sciences, National University, Los Angeles, CA.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-45982-9 (cloth)
1. Ordered algebraic structures. I. Kurdachenko, Leonid A., 1949- author. II. Subbotin, Igor Ya., 1950- author. III. Title.
QA172.D59 2015
511.3′3–dc23
2014022297
Abstract algebra is an essential part of a mathematics program at any university. It would not be an exaggeration to say that this area is one of the most challenging and sophisticated parts of such a program. It requires beginners to establish and develop a totally different way of thinking from their previous mathematical experience. Actually, to students, this is a new language that has proved to be very effective in the investigation and description of the most important natural and mathematical laws. The transition from the well-understood ideas of Calculus, aided by its many visual examples, to the abstraction of algebra, less supported by intuition, is perhaps one of the major obstacles that students of mathematics need to overcome. Under these circumstances, it is imperative for students to have a reader-friendly introductory textbook consisting of clearly and carefully explained theoretical topics that are essential for algebra and accompanied with thoughtfully selected examples and exercises.
Abstract algebra was, until fairly recently, studied for its own sake and because it helped solve a range of mathematical questions of interest to mathematicians. It was the province of pure mathematicians. However, much of the rise of information technology and the accompanying need for computer security has its basis in abstract algebra, which in turn has ignited interest in this area. Abstract algebra is also of interest to physicists, chemists, and other scientists. There are even applications of abstract algebra to music theory. Additionally, many future high school teachers now need to have some familiarity with higher level mathematics. There is therefore a need for a growing body of undergraduate students to have some knowledge of this beautiful subject.
We have tried to write a book that is appropriate for typical students in computer science, mathematics, mathematics education, and other disciplines. Such students should already possess a certain degree of general mathematical knowledge pertaining to typical average students at this stage. Ideally, such students should already have had a mathematics course where they have themselves written some proofs and also worked with matrices. However, the main idea of our book is that it should be as user-friendly to a beginner as possible, and for this reason, we have included material about matrices, mathematical induction, functions, and other such topics. We expect the book to be of interest not only to mathematics majors, but also to anyone who would like to learn the basic topics of modern algebra. Undergraduate students who need to take an introductory abstract algebra course will find this book very handy. We have made every effort to make the book as simple, understandable, and concise as possible, while leaving room for rigorous mathematical proofs. We illustrate the theory with a variety of examples that appeal to the previous experience of readers, which is useful in the development of an intuitive algebraic way of thinking. We cover only essential topics from the algebra curriculum typical for introductory abstract algebra courses in American universities. Through some of the numerous exercises, we introduce readers to more complex topics.
The book consists of five chapters. We start our exposition with the elements of set theory, functions, and matrix theory. In Chapter 2, we cover the main properties of the integers, viewed from an algebraic point of view. This paves the way for the final three chapters, covering Groups in Chapter 3, Rings in Chapter 4, and Fields in Chapter 5. These chapters cover the main beginning ideas of abstract algebra as well as sophisticated ideas. The book is accompanied by an Instructor’s solutions manual containing solutions for all exercises in the book.
The authors would like to extend their sincere appreciation to the University of Alabama (Tuscaloosa, USA), National Dnepropetrovsk University (Dnepropetrovsk, Ukraine), and National University (Los Angeles, USA) for their great support of the authors’ work. The authors also would like thank their family members for their patience, understanding, and much needed support while this work was in progress.
MARTYN R. DIXON
LEONID A. KURDACHENKO
IGOR YA. SUBBOTIN