Basic Algebra I
By addebook • Jul 14th, 2008 • Category: Mathematics
Basic Algebra I
Basic Algebra I
By Nathan Jacobson
Publisher: W.H.Freeman & Co Ltd
Number Of Pages: 472
Publication Date: 1974-01-01
ISBN-10 / ASIN: 0716704536
ISBN-13 / EAN: 9780716704539
Binding: Hardcover
Summary: the best intro to algebra for future mathematicians
Rating: 5
This book is by an expert algebraist who has rewritten his earlier introduction to algebra from the experience gained after 20 years as a Yale professor. It contains correct insightful proofs, carefully explained as clearly as possible without compromising their goal of reaching the bottom of each topic.
Other books say that one cannot square the circle with ruler and compass because it would require solving an algebraic equation with rational coefficients whose root is pi, and after all pi is a transcendental number. But Jacobson also proves that pi is a transcendental number, so as not to leave a logical gap. Naturally the burden on the student is somewhat higher than if he is merely told this fact without proof.
It is true that some other books include many more examples, and discuss them at extreme length, whereas Jacobson’s book is less than 500 pages, hence cannot include as many words. But Jacobson’s words are sometimes far better chosen, as he clearly understands the material at greater depth than other authors.
In his introduction to R modules, he discusses the most natural possible ring that acts on an abelian group: the ring of its endomorphisms. This is the true motivation behind the usefulness of R modules structures but is not even hinted at in most other books.
In his treatment of factorization in Noetherian domains, Jacobson carefully proves the existence of a single irredudible factor before proving existence of a complete factorization, thus avoiding perfectly a logical trap that some authors do not even notice.
In his discussion of the structure theory of finitely generated modules over a pid, he gives the concrete proof using diagonalization of matrices, that will actually be applied later to linear transformations, rather than some abstract existence proof that will be useless later, as many other authors do.
This sort of careful attention to the internal structure of the subject, and expert skill at presenting it correctly and clearly, are possible only to someone like Jacobson who is a true master of his area. I have only recently, as a mature mathematician, become aware of how wonderful his book really is for beginners who want to learn the subject correctly, from the beginning.
Some students not used to reading paragraphs, have been frustrated at his style of presentation, without realizing the superiority of his content. I can only recommend that those readers try harder to read his book, as it will repay far more than other sources.
Jacobson has made a sincere, and I think very successful effort, to write his 2 volumes on 2 different levels of sophistication, the first being back – bendingly clear and painstakingly organized as to the true logic of the subject.
After choosing a different source for my beginning graduate algebra course, I discovered the superiority of Jacobson, and wondered in amazement how such a great work could have been allowed to go out of print. After reading these reviews I understand. The readers who criticize the experts have eventually managed to veto the use of their works in classes. This makes the market share fall, and the books cease to exist. We have been obliged recently to remove Jacobson from our list of PhD references, in spite of its excellence, because it is out of print. This is a real disservice to our PhD students seeking to understand the material they will need to use.
Average students, i.e. most of us, have the right to learn a subject, but we should not have the right, and we are unwise to try, to vote the best books out of existence simply because we cannot understand them. Let us aspire to understanding the deeper treatment in Jacobson’s book. Let’s put our copy of Jacobson away and save it, if we cannot yet read it.
Clearly it is not the first book for everyone, but it is still perhaps the best, treatment of the material in existence to my knowledge at the upper undergraduate – graduate level, for the student who aspires to real mastery and understanding. If you want to be mathematician, you should get and read this book above others. Indeed the AMA rates both volumes of Jacobson as “essential” for every undergraduate library.
Summary: good textbook. but….
Rating: 3
During the lecture on the abstract algebra 1, Jacobson’s book was the main textbook. Although his book is regarded as “bible” by some professors, I cannot help confessing thath i need another reference book which explains the material in more detail and more carefully. So I consult Frayleigh’s algebra textbook whenever I hit upon something difficult to understand on jacobson’s. Frayleigh’s book is kinder, not to economize explanations and examples. Jacobson is one of the top-class mathematian, but his fame doesn’t assure his book’s efficiency. His book is written too abstractively. I don’t want to recommend this book to the beginers although it has the title “basic”.
Summary: Excellent!
Rating: 5
I’m truly shocked that this book has received such low reviews. This is not an easy book, but it shouldn’t be faulted for that. It’s clear and beautifully written, and it’s been a pleasure to work through. Additionally, the chapters are divided into sections that are ‘bite size’ with exercises at the end of each, which has made it well suited for regular daily study. I would highly recommend it to any student with some mathematical maturity who wishes to get a good foundation in the subject.
Summary: Superb book
Rating: 5
This book and its sequel BAII form a superb algebra resource that I use constantly. While this book is neither a reference (in the sense of Bourbaki) nor a textbook (its style is far too elegant to be classified as a textbook), it is beautifully written and one can learn a great deal by reading it. A word of warning though: this book presupposes a fair amount of mathematical maturity, so I would not recommend this book as an introduction to abstract algebra. On the other hand, it is a great complement to algebra courses and its originality and the variety of topics covered make it an invaluable resource.
Summary: Very confusing most of the time
Rating: 2
I dislike the book for two main reason:
1)the author hides information in essay-style paragraphs which usually end in phrases like, “thus, we have proved…”. If you are lucky, he will somewhere state the theorem.
2) because of 1) it serves as a poor reference book, which is the main subsequent value of a text after you have taken the course with it.
I do share some of the enthusiasm that Richard has for the book, but only for those parts of the book that I already understood from previous work. We actually took the class together at UC Santa Barbara. And from what I remember, Richard was one of the few students who had anything good to say about the book. I think a better graduate text is Hungerford’s or Grove’s (”Algebra”).
Free download Links
http://mihd.net/v18gyfp/
http://rapidshare.com/files/111117158/BAI_0_4.zip
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