“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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Adding in some applications may be good, too; I do not remember many being in Mac Lane. For over twenty years this text has been the “classic” work in its field.
When I taught “modern algebra” in “Math 6” the first time, inI began with sets and ended with groups. While it can be used as a reference, it should rather be read through carefully over a period of years – one must think in terms of birkohff if one wishes to absorb fully the algfbra and to do the problems.
Many algebra textbooks are so concerned about the process of learning to prove things that they communicate a sense of the subject as forbidding and stiff, dedicated to formalism and precision. The familiar domain of integers and the rational field are emphasized, together with the rings of integers modulo n and associated polynomial rings.
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.
Mac Lane has an enduring reputation as a great expositor. Chapter 11 includes a completely revised introduction to Boolean algebra and lattice theory. The ratio of definitions to theorems and exercises is kept low.
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 it.
This book is distinguished also by the great clarity with which all details have been presented. There are still many overqualified students and many underqualified students, and I can provide a constructive proof of that fact.
Survey of Modern Algebra
Chapters give an introduction to the theory of linear and polynomial equations in commutative rings. It also introduces the student to modules, but it biekhoff 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. It could probably work for teaching a first course to students who have never seen abstract algebra, if the instructor is at ease with it.
Byrelatively new concepts inspired by it had begun to influence homology theory, operator theory, the theory of topological groups, and many other domains of mathematics. The Mathemagician 2 6. Lots of people cannot birkhofr Lang’s writing, too. Vinberg seems similar but more intense, so I imagine it would work well, too.
It is a good, once standard book, but there are texts that better suit the needs of modern students. The classical algebra is nicely embedded in this structure, as are also applications to other fields of thought. Our Survey in presented an exciting mix of classical and conceptual ideas about algebra.
Motivation, examples, clear writing, reasonable exercises, they are all there. But they want to teach them algebra even more. We must not risk them never learning mavlane. The word “survey” in the title is quite accurate; for, although many topics are treated, none of them is really completely developed.
These new directions involve algebraic geometry, computers, homology, and representations A Survey of Modern Algebra has been rewritten as Mac Lane-Birkhoff, AlgebraMacmillan, New York,and this version introduces categorical methods; category theory emerged from algebraic topology, but was then used by Grothendieck to revolutionize algebraic geometry.
It is a great supplementary text, however. Getting across the rich motivation, ideas, analogies, history, and so forth should be an explicit goal, but it will not always be possible. The revised edition differs only in minor rearrangements and additions.
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 to work without much issue.
Only in this way will they be able to appreciate the full richness of the subject. One of the best things about this book is the balanced approach to rigour and abstractness in relation to intuitive appreciation and concrete application. I think you could cover the entire book, minus perhaps Chapter XIV on lattices, which are not typically emphasized and are partly historical, and possibly Chapter VIII, which is a bit odd, in my opinion.
Here care is taken to keep in the foreground the fundamental role played by algebra in Euclidean, affine, and projective geometry. Maximum generality entailing sophisticated machinery can seem efficient in the abstract, but it rarely works so well in practice. The most striking characteristic of modern algebra is the deduction of the theoretical properties of such formal systems as groups, rings, fields, and vector spaces.
The next year Mac Lane put group theory first, and set theory Boolean algebra last! A semester course on abstract algebra could deal with Chapters, 11, 13, and For undergraduates, we obviously need to ensure they see examples, learn the basic theory, be able to write proofs, etc. I would certainly place it in the hands of an interested undergraduate wondering what algebra was all about, particularly one who had already taken linear algebra.
The longer chapters are his; the shorter ones mine. 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.
This seems especially important in an elementary text because it serves to emphasize the fact that the abstract concepts all arise from the analysis of concrete situations. You mention Herstein and Vinberg, which are good; Artin or Gallian also work, for example. It has a lot of linear algebra, which is good, and it is not too hard, but it requires some work.
These ideas are still most relevant and worthy of enthusiastic presentation.
Chapter IIIand some field and vector space theory c. Rotman’s Advanced Modern Algebra is pretty light on the Galois theory but he gets points for having a chapter on algebras. I think using many texts birkhogf often and in this case almost certainly a good idea. Modern algebra also enables one to reinterpret the results of classical algebra, giving them far greater unity and generality.
Numerous additional exercises, summarising useful formulas and facts, have been included. Chapter 6 introduces noncommutative algebra through its simplest and most fundamental concept: Such students are strongly advised to do supplementary reading, or at least browsing, in the references listed at the end.
Modern algebra prospered mightily in the decadesfrom functional analysis to algebraic geometry – not to mention our own respective researches on lattices and on categories.
I’m working on such a book myself,but for low cost texts only.