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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Then the abstract definition appears simple, and the theoretical properties which are deduced from the definition exhibit the power of the concept. Also,the author writes very well indeed.
We have done this by illustrating each new term by as many familiar examples as possible. While I’m really enjoying the book,I’ve got a real dilemma gnawing at me. 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.
We have also tried not to lose sight of the fact that, for many students, the value of algebra lies in its applications to other fields: Such students are strongly advised to do supplementary reading, or at least browsing, in the references listed at the end. 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.
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 wlgebra twentieth century mathematics. Beyond this, occasional sections have been revised and a few problems have been added to some of the exercises.
It is a unified and comprehensive introduction to modern algebra. Vinberg seems similar but more intense, so I imagine it would work well, too. 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. Although two or three books on the new algebra have already appeared in English, the present volume appears to the reviewer to be the best maclanw introduction to the subject, unique in its clarity, balance, generality and inclusiveness.
The authors are quick to indicate applications and careful to motivate and illustrate abstractions. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new. If you alter the presentation maclan your lectures to skew more toward examples and concrete proofs, ideally while still discussing the more abstract stuff a bit, then I think this book can work well. The original comprehensive Survey has been reordered somewhat and augmented to the extent of approximately fifty pages.
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. Dual spaces and tensor products are also discussed, but generalizations to modules algera rings are not considered.
Algebra – Saunders Mac Lane, Garrett Birkhoff – Google Books
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. 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 of group theory, then modules and advanced linear algebra, followed by field and Galois theory, representation 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 the end, if possible e.
The familiar domain of integers and the rational field are emphasized, together with the rings of integers modulo n and associated polynomial rings.
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. A ‘Survey of Modern Algebra’ made it possible to teach an undergraduate course that reflected the richness, vigour, and unity of the subject as it is growing today.
That was characteristic of his freshness, his initiative, and his lack of respect for conformity; but it came as a slight shock to me at the time. I had taught algebra courses at Harvard when I was an instructor, and at Cornell I taught algebra out of the book by Bocher; at Chicago, out of a book, ‘Modern Higher Algebra’ by Albert; and at Harvard again out of my own notes.
At both the graduate and undergraduate level, it seems there are books that are better in parts or in whole. I have come to like Gallian quite a bit since first giving it a real chance to see what my school teaches.
Only in this way will they be able to appreciate the full richness of the subject. I think graduate courses should use category theory pretty openly.
Email Required, but never shown. Or, maybe you have a truly prodigious bunch who have all seen algebra before or who are as mature as most incoming graduate students at good schools, in which case following the text more closely ough virkhoff work without much issue.
Much of this significance, it seems to us, lies in the imaginative appeal of the subject. Mac Lane was elected to the National Academy of Sciences in 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.
They embody the elegance, precision, and generality which are the hallmark of mathematics!
There is also contact with the field of mathematical algerba in the chapter on the algebra of classes and with the ideas of topology in the proof of the fundamental theorem of algebra. It also introduces the student to modules, but it does not insist on working with modules instead of vector spaces whenever possible, which is probably good, because modules often serve to slightly confuse without adding anything more than a bit of generality.
For example, a semester or quarter course covering linear algebra may be based on Chaptersthe real and complex fields being emphasized. Birkhofff authors express the belief that “for many students, the value of algebra lies in its applications to other fields: 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 birkhofc, which is usually what algebra professors focus on in a first graduate course.
Birkhoff and Mac Lane’s book first appeared inand van der Bitkhoff book first appeared in Post as a guest Name.
In both undergraduate courses and graduate courses students’ abilities and background are quite variable; that much has not changed in the past few decades. 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.
Although not enough to reconcile this problem, their exposition on categorical notions is quite clear, at least, albeit maybe not as good as some treatments that have since come. I feel there is a need in mathematics for detailed discussion of available resources.