“Algebra” by MacLane and Birkhoff was the first abstract algebra book I bought and I loved it! It wasn’t the first abstract algebra book I read. This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary. Documents Similar To Saunders Mac Lane, Garret Birkhoff Algebra Mathematics Form and Function. Uploaded by. jacinto [Charles C. Pinter] a Book.
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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. Also, your review page is quite comprehensive! Aluffi 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.
There are contacts with many branches of mathematics and so it can serve as an introduction to nearly the whole of modern mathematics. If you are looking for strengthening your intuition, please look into Gallian’s book “Contemporary Abstract Algebra”. 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 how often they will find themselves ignorant of those things.
Would it be suitable for self-study? 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 algebraa ‘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. My recommendation is to start with a different book I like Dummit and Foote.
Being able to handle abstraction does not mean students should not learn many concrete, basic examples, however, nor does it mean alggebra should learn things at the most abstract possible level and be expected to figure out the less abstract birkyoff on their own.
Hungerford is a common alternative I did not discuss, but I have never found it very exciting. While I’m really enjoying the book,I’ve got a real dilemma gnawing at me.
Sign up or log in Sign up using Google. Getting across the rich motivation, ideas, analogies, history, and so forth should be an explicit goal, but it will not always be possible. In this way, I do not expect today’s students are substantially better than past students, either.
This is not to say Mac Lane and Birkhoff do this! I think these principles in lecture and text, along with hard work by the students, lots of student-student communication including some forced working togethersome student-professor communication including some forced communication, e.
This book is distinguished by its procedure from the concrete to the abstract. The style in which the book is written is like a story. Chapter 11 includes a completely revised introduction to Boolean algebra and lattice theory.
Exercises of the latter type serve the important function of familiarizing the student with the construction birlhoff a formal proof. Just a few questions though: Perhaps a lecture reviewing elementary set theoretical notions, then cover some linear algebra introducing basic category theory after seeing albebra 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 birkhocf and the likeand finally homological algebra, with some advanced or extra topics at the end, if possible e.
Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: 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.
A semester course on abstract algebra could deal with Chapters, 11, 13, and He returned to Harvard inthe year after I had given a course in modern algebra on the undergraduate level for the first time. Here care is taken to keep in the foreground the fundamental role played by algebra in Euclidean, affine, and projective geometry. Our third edition, infinally included tensor products of vector spaces, while the fourth edition clarified the treatment of Boolean algebras and lattices.
The next year Mac Lane put group theory first, and set theory Boolean algebra last! In this book Professors Birkhoff and Mac Lane have made an important contribution to the pedagogy of algebra. Instructors who have used the original edition with college classes appreciate its scope.
Therefore, instead of omitting these results, we have attempted to incorporate macane systematically within the framework of the ideas of modern algebra. During the s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane, substantially advanced American teaching and research in abstract algebra.
Surely, people appreciate the extant ones e.
The short version is I think the book can be used for either undergraduates or graduates with some success, but I think macane is less than ideal in both cases. Their emphasis is on the methods and spirit of modern algebra rather than on the subject matter for itself. 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 madlane assign boring lemmas and the like as exercises or reading but only truly easy ones as exercises!
There should also be specific portions of book and lecture that are intentionally plain, definition-theorem-proof type things, but where all the richness maaclane introduced later down the line, as to not snub the students.
Algebra – Saunders Mac Lane, Garrett Birkhoff – Google Books
These ideas are still most relevant and worthy of enthusiastic presentation. Yet, I also think a course should start with basic material. Chapter VIImore field theory and macalne algebra c.
The authors are to be congratulated on having improved an already excellent text. The Mathemagician 2 6. It wasn’t the first abstract algebra book I read, though, but it probably was the first book that I read carefully.
Some material, especially that on linear algebra, has been rearranged in the light of experience. Preferably one that does not use category theory, unless you are a genius this will probably save you from a lot of suffering.
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. Our Survey presented an exciting mix of classical, axiomatic, and conceptual ideas about algebra at a time when this combination was new. Birkhoff and Mac Lane’s book first appeared inand van der Waerden’s book first appeared in I altebra not believe students are significantly less capable today than they were several decades ago, as you seem to suggest.