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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Survey of Modern Algebra
We also hope it will give a solid introduction to this fascinating and rapidly growing subject, to those students interested in modern algebra for its own sake. The original comprehensive Survey has been reordered somewhat and augmented to the extent of approximately fifty pages.
Usually,for strong undergraduate courses, he’d supplement Algebra with examples drawn from he and Birkoff’s classic undergraduate text, A Survey Of Modern Algebra. This is one of the primary reasons an undergraduate course ought algwbra supplement this book. Hungerford’s Algebra is a pretty good book, but the author includes little about homological algebra, and the only time you see the word “representation” is when discussing category theory.
The ratio of definitions to theorems and exercises is kept low. In the new third edition, the authors have modernized and improved the material in many details.
Although two or three biekhoff on the new algebra have already appeared in English, the present volume appears to the reviewer to be the best all-round introduction to the subject, unique in birkoff clarity, balance, generality and inclusiveness.
This has influenced us in our emphasis on the real and complex fields, on groups of transformations as contrasted with abstract groups, on symmetric matrices and reduction to diagonal form, on the classification of quadratic forms under the orthogonal and Euclidean groups, and finally, in the inclusion of Boolean algebra, lattice theory, and transfinite numbers, all of which are important in mathematical logic and in the modern theory of real functions.
I think the needs of the graduate student are more complex, in a sense, than the undergraduate’s needs, because many are less obvious than in the undergraduate case.
Rotman may be a good primary text. I quite brkhoff Etingof’s Introduction to representation theorywhich is available at his sitebut it takes two chapters before it gets to the case of finite groups, which is usually what mzclane professors focus on in a first graduate course. I love Lang, especially for things like Galois theory, but it is too hard, too fast, too big, too encyclopedic, and, dare I say, too modern for most graduate students.
Only the last five chapters remain unchanged. I do algfbra think using algebras is a problem, but birmhoff whole chapters is probably a bit much, so some would have to be cut. It is unlikely you will cover the entire book, either, even in a one year course. This independence is intended to make the book useful not only for a full-year course, assuming only high-school algebra, but also for various shorter courses.
Chapter VIImore field theory and linear algebra c. Finally, the last three chapters provide an introduction to general commutative algebra and arithmetic: Adopting this as the main textbook for an undergraduate abstract algebra course would today be an eccentric move. Bigkhoff is really something special, but it is probably a bit too category theory-centric for students new to the subject in the sense that they will not have enough familiarity with the lower-level stuff.
I think graduate courses should use category theory pretty openly.
There are other fairly popular graduate algebra texts that do not cover all the presumed topics, however. 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.
These theorems are then applied to some familiar maclqne to some less familiar examples, thus broadening the student’s viewpoint without getting him lost in abstractions. A typical first undergraduate course may cover group theory through the isomorphism theorems and the structure theorem for finite abelian groups,possibly including group actions and the Sylow theorems c.
A second course may cover a bit more group theory c. It provided a synthesis formerly obtainable only after much more advanced work. It was the standard textbook for undergraduate courses in modern algebra.
It is a great supplementary text, however. One of us would draft a chapter and the other would revise it. I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and algbera partly historical, and possibly Chapter VIII, which is a bit algebfa, in my opinion.