Basic Algebra I: Second Edition [Paperback]

I mean it: 5 stars -- I can think of few books I'd rate as highly. But one warning: This book suits my taste for essay-like exposition, an approach diametrically opposed to the more common practice of tables of numbered theorems and symbol-only proofs. If you prefer to think of a module as a homomorphism from a ring into the ring of endomorphisms of an Abelian group, this book is the right approach; if you prefer a list of equations defining a module, then do not use this book as your primary source. Few mathematicians have a good sense of language, but the comma splices, absent articles, poor syntax, near-aphasia, and sentence-fear prevalent in many texts are absent here. If you are writing for human beings, your text need not obey Fortran syntax, as does Hungerford's. Jacobson can write. In English.

The chapter on Galois theory covers more finite Galois theory than most algebraists need to know. Jacobson's style of merging the necessary symbolism into an essay-like presentation is strongest in this long chapter, and planted concepts firmly in my mind in the same language that I would use to describe them in conversation.

But mathematics is not learned through conversation, and that is the crux of the objections to this book. I'm grateful that my undergraduate professor used this book, but I would not recommend doing so, as some good students do not read it easily. I'd use this as a secondary text, with Hungerford, Fraleigh, or Herstein as the primary source. Mathematical writing would be better if all students saw Jacobson's approach at some point in their careers, Jacobson is the best exponent of terse, clean, textbook-as-essay style. Should you dislike his approach, exposure to his style might still broaden your ideas on mathematical exposition and help you better define your own style; a reaction against his methods can sharpen your own game.

A masterful book.

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.

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.

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