A survey of modern algebra. Bibliography: p. Includes index. 1. Algebra, Abstract. I. MacLane, Saunders, (date) joint author. II. Title. QAB57 “This classic text introduces abstract algebra using familiar and concrete examples that illustrate each concept as it is presented. It covers such topics as the role. A survey of modern algebra / by Garrett Birkhoff and Saunders MacLane Birkhoff, articles: Garrett Birkhoff, Greatest common divisor, Saunders Mac Lane.
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He returned to Harvard inthe year after I had given a course in modern algebra on the undergraduate level for the first time. Then birkhocf later we took part in the flow of new ideas from discovery to use and to present to students.
Much of this significance, it seems to us, lies in the imaginative appeal of the subject.
The selection of exercises is sufficient to allow an instructor to adapt the text to students of quite varied degrees of maturity, of undergraduate or first year graduate level. Accordingly, we have tried throughout to express the conceptual background of the various modrn used. To develop the student’s power to think for himself in terms of the new concepts, we have included a wide variety of exercises on each topic.
Byrelatively new concepts inspired by it had begun to influence homology theory, operator theory, the theory of topological groups, and many other domains of mathematics. The most striking characteristic of modern algebra is the deduction of the theoretical properties of such formal systems as groups, rings, fields, algebda vector spaces. We aimed to combine the abstract ideas with suitable emphasis on examples and illustrations. Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept.
All instructor resources are now available on our Instructor Hub. While it can be used as a reference, it should rather be read through carefully over a period of years – one must think in terms of years if one wishes to absorb fully the material and to do the problems.
Probably the best way to appreciate the vitality and growth of mathematics today is to study modern algebra. Chapters give an introduction to the theory of linear and polynomial equations in commutative rings. A famous mathematician once remarked to me that everyone he knew who had worked through A Survey of Modern Algebra had come to love the subject.
Throughout the study of matrices and quadratic forms the geometric point of view is emphasized. Although many of the basic results stem back to Kronecker, Dedekind and Steinitz, the present-day subject is largely the creation of the great woman mathematician, Emmy Noether.
This independence is intended to make the book useful not only xnd a full-year course, assuming only high-school algebra, but also for various shorter courses.
A Survey of Modern Algebra
Saunders Mac Lane was the author or co-author of more than research papers and six books. Although some additions and rearrangements have been made for this edition, the content remains essentially the same [as the edition]. He received the nation’s highest award for scientific achievement, the National Medal of Science, in Nowhere can teachers better catch today’s spirit of mathematics.
Interesting historical references appear in a number of places. It does this by discussing examples of mathematical systems or situations already partially familiar to the student, isolating important properties of these as postulates, and deducing some of the consequences of these postulates.
Chapter 11 includes a completely revised introduction to Boolean algebra and lattice theory.
Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new. The authors express the belief that “for many students, the value of algebra lies in its applications to other fields: The revised edition differs only in minor rearrangements and additions.
Here care is taken to keep in the foreground the fundamental role played by algebra in Euclidean, affine, and projective geometry. I had taught out of most of the extant books. It could be through conference attendance, group discussion or directed reading to name just a few examples.
Mac Lane had had much more teaching experience than I, and I think the popularity of our book owes more to him than to me. I would certainly place it in maclaje hands of an interested undergraduate wondering what algebra was all about, particularly one who had already taken linear algebra.
Survey of Modern Algebra
Instructors who have used the original edition with college classes appreciate its scope. Beyond this, occasional sections have been revised and a few problems have been added to some of the exercises.
We also hope it will give a solid introduction to this fascinating and rapidly growing subject, to those students interested in modern algebra for maclanne own sake. Garrett Birkhoff published more than papers and supervised more than 50 Ph.
Garrett and I combined our preliminary notes to publish with MacMillan in our joint book, Survey of Modern Algebra. Therefore, instead of mcalane these results, we have attempted to incorporate them systematically within the framework of the ideas of modern algebra.