Preface. Birkhoff & Mac Lane’s Algebra is a brilliant book. I should probably spend some time with it again, actually. Also, I apologize for such a. In Garrett Birkhoff and Saunders Mac Lane published A Survey of Modern Algebra. The book became a classic undergraduate text. Below we examine a. Garrett BirkhoffHarvard University Saunders Mac Lane The University of Chicago A SURVEY OF ern fourth.
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It is unlikely you will cover the entire book, either, even in a one year course. Numerous additional exercises, summarising useful formulas and facts, have been included. Only in this way will they be able to appreciate the full richness of the subject.
Perhaps a lecture reviewing elementary set theoretical notions, then cover some linear algebra introducing basic category theory after seeing direct sumsthen cover some ring theory, then plenty kaclane group theory, then modules and advanced linear algebra, followed by field and Galois theory, girkhoff theory using algebras and specializing quickly to groupscommutative algebra including some applications to algebraic geometry and the likeand finally homological algebra, with some advanced or extra topics at birkhpff end, if possible e.
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. Post as a guest Name. Artin 2e, not 1e is also good and emphasizes linear algebra and geometric intuition, which is good considering how often students will need things from linear algebra and maclanne often they will find themselves ignorant of those things.
But, hey, at least there are lattices!
But I’d sure keep one on reserve in the library for my students to browse. I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and are partly historical, and possibly Chapter VIII, which is a algehra odd, in my opinion. The revised edition differs only in minor rearrangements and additions.
But there is no dearth of good reference works in algebra, and in the reviewer’s opinion the present textbook will prove more useful than another encyclopedic treatise would have been. Algebra was almost unheard of at the undergraduate level not too long ago, whereas now an undergraduate program would be remiss not to biirkhoff students take at least an introductory course covering the basic theories of groups, rings, and fields.
Judson is good all around.
Some combination of the approaches of the above books ought to make an excellent undergraduate course. Moreover, it was written in a clear and enthusiastic style that conveyed to the reader an appreciation of the aesthetic character of the subject as well as its rigour and power.
This is one of the primary reasons an undergraduate course ought to supplement this book. Adopting this as the main textbook for an undergraduate abstract algebra course would today be an eccentric move. I don’t think hitting students with category theory unless they’ve already had considerable exposure to algebra is a good idea. For undergraduates, we obviously need to ensure they see examples, learn the basic theory, be able to write proofs, etc.
Chapter VIImore field theory and linear algebra c.
Potential Courses I think the text could be used for either undergraduate or graduate students with decent results, but I do not think it is the best choice as a primary text. These new directions involve algebraic geometry, computers, homology, and representations A Survey of Modern Algebra has been rewritten as Mac Lane-Birkhoff, AlgebraMacmillan, New York,and this version introduces categorical methods; category theory emerged from algebraic topology, but was then used by Grothendieck to revolutionize algebraic geometry.
But at the same time, since the book contains very little homological algebra and no representation theory beyond the definition,can it really be used as graduate course today?
It is a great supplementary text, however. Because there is quite a bit, and because students are assumed to be mature and to be getting used to reading math, not just going to lecture, one can easily assign boring lemmas and the like as exercises or reading but only truly easy ones as exercises!
Chapter 11 includes a completely revised introduction to Boolean algebra and lattice theory. We had tried out our somewhat differing ideas of how this should be done in a course at Harvard for three successive years, before reorganizing and presenting them in textbook form. In writing the present text we have endeavoured to set forth this formal or “abstract” approach, but we have been guided by a much broader interpretation of the significance of modern algebra.
Students of the first type would benefit from a more challenging text, whereas students of the second type would benefit from an easier one. Yet, I also think a course should start with basic material. His paper, “On the Structure of Abstract Algebras” founded a new branch of mathematics, universal algebra.
I’m going to soon review the 3rd edition of Saunders MacLane And Garrett Birkoff’s Algebra at my blog soon and this is the first time I’m really carefully reading it.
As aforementioned, some category theory, especially an awareness of universal properties and basic definitions e. The idea is to a. Certainly, algebraists expect everyone has seen some homological algebra and representation theory by the time they get their Ph.
This approach, which crystallized earlier insights of Cayley, Frobenius, Kronecker, and Dedekind, blossomed in Germany in the s. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new.
Algebra – Saunders Mac Lane, Garrett Birkhoff – Google Books
I often found it dry, too. The longer chapters are his; the shorter ones mine. It is a good, once standard book, but there are texts that better suit the needs of modern students.