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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The present edition represents a refinement of an already highly useful text. This approach, which crystallized earlier insights of Cayley, Frobenius, Kronecker, and Dedekind, blossomed in Germany in the s. He received the nation’s highest award for scientific achievement, the National Medal of Science, in It began to sell well as soon as the war was over, in at about – annually.
The book was prepared at a time when both of us were assistant professors, so without tenure. The most important parts of each theory are included and that is all that can be asked of an birkhotf textbook.
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 algevra. Students of the first type would benefit maclanr a more challenging text, whereas students of the second type would benefit from an easier one. We hope that the present book will prove an adequate reference for those wishing to apply the basic concepts of modern algebra to other branches of mathematics, physics, and statistics.
For a first year graduate course-which at Harvard,usually included a number of undergraduates-he’d use the book straight up and cover most of it. This book is distinguished also by the great clarity with which all details have been presented.
The truth, though, is that undergraduates are fairly unlikely to read their textbook. What kinds of students can this book be used for in ? In we prepared a carefully polished second edition, in birlhoff polynomials over general fields were treated before bir,hoff to the real field. Garrett Birkhoff published more than papers and supervised more than 50 Ph. I doubt they’ll be able to fully appreciate the enormous generalization and unification it provides without a giant stock of examples which is why I love Emily Riehl’s Category Theory In Context as the definitive introduction to category theory for mathematics students.
Mac Lane birkboff an enduring reputation as a great expositor. Birkhoff do not avoid using universal properties, and they do not always bother to give students something concrete to hold on. For undergraduates, we obviously need to ensure they see examples, learn the basic theory, be able to write proofs, etc. For the social scientist whose mathematical studies have reached through the calculus, this book can confidently be urged as the thing to study next.
After explaining the conceptual content of the classical theory of equations, our book tried to bring out the connections of newer algebraic concepts with geometry and analysis, connections that had indeed inspired many of these concepts in the first place.
In the new third edition, the authors have modernized and improved the material in many details. Chapter IIIand some field and vector space theory c. Our Survey in presented an exciting mix of classical and conceptual ideas about algebra. But the authors’ delight in what was then a new subject shines through their writing, and their willingness to be informal when necessary was a smart move.
The rejuvenation of algebra by the systematic use of the postulational method and the ideas and point of view of abstract group theory has been one of the crowning achievements of twentieth century mathematics. Indeed, I find his writing quite agreeable.
Because there is quite a bit, and because maclame 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!
There are other fairly popular graduate algebra texts that do birkuoff cover all the presumed topics, however. One of the best things about this book is the balanced approach to rigour and abstractness in relation to intuitive appreciation and concrete application.
Dual spaces and tensor products are also discussed, but generalizations to modules over rings are not considered. It is a great supplementary text, however. Those are perhaps worth substantial time, even if they are somewhat too odd for most courses. Ideally, the course book would have lots of this and the lecture somewhat less.
I do not think using algebras is a problem, but three whole chapters is probably a bit much, so some would have to be cut. That the basic objects mxclane groups wlgebra rings are presented early on and come back to several times, rather than spending one-hundred pages only talking about groups, then changing to only talking about rings, should also help students see how algebraic structures are related; it albebra makes the beginning easier, and provides an opportunity for students to save themselves if they fall behind early on.
The “Modem Algebra” of our title refers to the conceptual and axiomatic approach ad this subject initiated by David Hilbert a century ago.
The advantage of not falling back on a general reference as a supplement is you can use the really great parts of these excellent books, and the students know where to go for much more; plus, both of these are written at a reasonably low level. I don’t think hitting students with category theory unless they’ve already had considerable exposure to algebra is a good idea.
Survey of Modern Algebra
Their emphasis is on the methods and spirit of modern algebra rather than on the subject matter for itself. Although my course was well attended, I was much more research-oriented than teaching-oriented. This reviewer can testify to its appeal to students.
I think graduate courses should use category theory pretty openly.
We enjoyed teaching and writing algebra because it was clear, exciting, and fun to present. Instructors who have used the original edition with college classes appreciate its scope.