Introduction to abstract algebra. (Q2718879)

The first edition of this introductory textbook appeared as early as in 1960. Back then, written by Neal McCoy, this first course in abstract algebra was one of the very first textbooks of this kind, apart from the already existing, much more comprehensive classics on modern algebra by B. L. van der Waerden or G. Birkhoff-S. MacLane. In contrast to those voluminous, almost encyclopedic standard texts, McCoy's book was designed for undergraduate students (and their teachers) only. Its main goal was that of making the first steps into abstract algebra as simple and clear as possible while retaining the precision required to give the beginner an introduction to the fundamental ideas, methods, and results of abstract algebra. Following this prevailing methodical strategy just as consequently as masterly, Neal McCoy produced a text that became utmost popular, wide-spead and much used in the sequel, especially at American colleges and universities. As a consequence (and proof) of its popularity, the book has undergone five new editions between 1968 and 2001, with several revisions, up-datings, and improvements. The early editions were written by N. McCoy himself, whilst the later revisions were done in collaboration with G. Janusz.NEWLINENEWLINENEWLINEThe book under review is now the sixth edition of this popular evergreen. The text, rooted in the successful tradition of earlier editions, has been revised once more. This sixth edition contains many new exercises as well as a new section on applications to RSA cryptology. Two topics, order properties of integers and set-theoretic foundations, have been moved forward in the book so that they can be used more systematically in the proofs. Also, a number of additional comments, remarks, and exercises now point the reader toward more recent and advanced topics in algebra. These revisions have certainly improved, once more, the didactic value of this trusty first introduction to abstract algebra.NEWLINENEWLINENEWLINEOf course, one characteristic peculiarity of this book, which could be called its trade-mark, has been kept unchanged. Namely, the authors' approach of presenting the theory of rings in the beginning, and discussing groups only in the second half of the book, has been retained with good reasons. After all, both the teaching experience of the authors and the continuous success of the book hold good, now as before. As to the contents, the text consists in nine chapters, each of which is subdivided into several sections.NEWLINENEWLINENEWLINEChapter I provides a brief study of abstract rings, together with various examples of concrete rings.NEWLINENEWLINENEWLINEChapter II introduces ideals, ring homomorphisms, factor rings, and the isomorphism theorems. The ring of integers is characterized by its order properties.NEWLINENEWLINENEWLINEChapter III discusses integral domains and fields, with special emphasis on the fields of real and complex numbers.NEWLINENEWLINENEWLINEChapter IV deals with divisibility and factorization in rings on the elementary level, focusing on the ring of integers and its factor rings. This is applied in a brief new section on RSA cryptography.NEWLINENEWLINENEWLINEChapter V is devoted to polynomial rings of one variable over a field, including splitting fields and polynomials over the field of rational numbers.NEWLINENEWLINENEWLINEChapter VI turns then to groups and their general properties. Normal subgroups, factor groups, cyclic groups, the symmetric and the alternating group, and an application to the 15-puzzle as an illustration form the topics conveyed here.NEWLINENEWLINENEWLINEChapter VII explains the basics on finite abelian groups, including elementary divisors, the classification theorem, and (as an application) the structure of the multiplicative group of a finite field.NEWLINENEWLINENEWLINEChapter VIII turns to finite groups in general, focusing on Sylow's theorems and their applications.NEWLINENEWLINENEWLINEChapter IX adds some more topics on rings and fields. Expanding ideas and methods studied in the previous chapters, the authors discuss the factoriality of principal ideal domains, touch upon Euclidean domains and give a proof of the Fundamental Theorem of Algebra.NEWLINENEWLINENEWLINEThe text comes with a huge amount of exercises providing considerable additional information to the reader. Some exercises contain important results that go beyond the scope of the text material, whereas others simply reinforce the material presented in the particular section. A few new exercises have been added to present more challenge to the strong student. Numerous examples are scattered throughout the text and make the abstract concepts more perceptual and beneficial.NEWLINENEWLINENEWLINEAltogether, this popular classic has been made available again, in an enriched and improved form. The new generation of beginners in abstract algebra will profit from this tailor-made, didactically highly valuable introduction just as much as the generations before did, as one rare peculiarity of this book will be always appreciated: dedication to students and consideration of their particular needs as beginners. Besides, the text does not require any prerequisites, not even linear algebra or matrix calculus, and it is totally self-contained, therefore perfectly suited for private study or reading assignment, too.