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.
A survey of modern algebra / by Garrett Birkhoff and Saunders MacLane – Details – Trove
The longer chapters are his; the shorter ones mine. Accordingly, we have tried throughout to express the conceptual background of the various definitions used. What kinds of students can this book be used for in ? We hope that our book will continue to serve not only as a text but also as a convenient reference for bikhoff wishing to apply the basic concepts of modern algebra to other branches of mathematics, including statistics birkyoff computing, and also to physics, chemistry, and engineering.
I think it would definitely be too difficult for any but the very best undergraduates and I think there are several books that would be better for that purpose: Only in this way will they be able to appreciate the full richness of the subject.
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.
It’s a projected book to accompany my new publishing company to instruct penniless students on low-cost mathematics sources for self study. Interesting historical references appear in a number of places.
Email Required, but never shown. To develop the student’s power to think for himself in terms of the new concepts, we have included a wide variety of maclanr on each topic. It is a great supplementary text, however. 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. I am less sure what makes a really excellent graduate course in terms of extant texts.
Lots of people cannot stand Lang’s writing, too.
Algebra – Saunders Mac Lane, Garrett Birkhoff – Google Books
Ideally, the course book would have lots of this and the lecture somewhat less. I quite like 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 algebra professors focus on in a first graduate course.
Finally, the last three chapters provide an introduction to general commutative algebra and arithmetic: 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. Both are excellent books Ibrkhoff have called this book Advanced Modern Algebra in homage to thembut times have changed since their first appearance: 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.
One of us would draft a mwclane and the other would revise it.
Preliminary Thoughts In both undergraduate courses and graduate courses students’ abilities and background are quite variable; that much has not changed in the past few decades. The authors express the belief that “for many students, the value of algebra lies in its applications to other fields: 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.
Our book, first published 50 years ago, was intended to present this exciting new view of algebra to American undergraduate and beginning graduate students. Our Survey in presented an exciting mix of classical and conceptual ideas about algebra. A very striking feature of the book is its broad point of view.
Birkhoff and Mac Lane also want to teach their students to prove things, of course. Thus there is a careful development of real numbers, such as Dedekind cuts, and such set-theoretic concepts as order, countability and cardinal number are discussed. I understand from professors who were students at Harvard that MacLane himself would use the book-in all it’s versions,from the first mimeographed drafts in the early ‘s to the 3rd edition he used in his last teaching days in the mid’s-for both undergraduate and graduate courses in algebra depending on the strength of the students, which would vary enormously from class to class.
The “Modem Algebra” of our title refers to the conceptual and axiomatic approach to this subject initiated by David Hilbert a century ago. Because of it’s unorthodox structure and old fashioned style,though,I’m not sure I’d use the book for a graduate course.
A section on bilinear forms and tensor products has been added to the chapter 7 on vector spaces, while Chapter 11, now entitled “Boolean algebras and birkhoff, contains a new introduction to Boolean algebras, as well as a section on the representation of such by sets.
Unfortunately, I am not sure what the best supplements are. In both undergraduate courses and graduate courses students’ abilities and background are quite variable; that much has not changed in the past few decades. That may overstate things, since my friend probably knows more mathematicians than students who got fed up and left. But at the same time, since the book contains very little homological algebra and no representation theory beyond the definition,can it bikrhoff be used as graduate course today?
This reviewer can testify to its appeal to students. We enjoyed teaching and writing algebra because it was clear, exciting, and fun to present.
I macclane come to like Gallian quite a bit since first giving it a real chance to see what my school teaches. 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.
The Basic Graduate Year includes some good applications to algebraic geometry and algebraic number theory, for example. Their emphasis is on the methods and spirit of modern algebra rather than on the subject matter for itself. We have websites like Math Stack Exchange where we can ask users for opinions, but it would be nice to have many more bibliographies of mathematical subjects than there currently are. He returned to Harvard inthe year after I had given a course in modern algebra on the undergraduate level for the first time.
It is unlikely you will cover the entire book, either, even in a one year course.
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. You do not define a group as a groupoid with one object, at least not maclnae an introductory course.