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Basic Algebra I: Second Edition (Dover Books on Mathematics) 2nd Edition

3.5 out of 5 stars 27 customer reviews
ISBN-13: 978-0486471891
ISBN-10: 0486471896
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 528 pages
  • Publisher: Dover Publications; 2 edition (June 22, 2009)
  • Language: English
  • ISBN-10: 0486471896
  • ISBN-13: 978-0486471891
  • Product Dimensions: 6.1 x 1.1 x 9.2 inches
  • Shipping Weight: 1.6 pounds (View shipping rates and policies)
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (27 customer reviews)
  • Amazon Best Sellers Rank: #177,244 in Books (See Top 100 in Books)

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Customer Reviews

Top Customer Reviews

Format: Hardcover
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.
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Format: Hardcover
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.
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By Remi on August 7, 2002
Format: Hardcover
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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Format: Hardcover
Jacobson's choice of topics is interesting; he does not simply stick to the canonical list of topics that the typical "first course in algebra" book covers, but touches on things like Lie and Jordan algebras, lattices, and the decidability of the first-order theory of the reals. For a budding research mathematician, this is great, since these are currently active areas of research. But for the average student, I would recommend something else. Several important theorems are not labelled theorems or given theorem numbers, e.g., the class equation for a finite group, or the fact that a polynomial equation is solvable in radicals if and only if its Galois group is solvable. That principal ideal domains are assumed to be commutative is not explicitly stated, and the notation "E/F" for "E, a field extension of F" is not explained. The Jordan-Holder theorem is not in the chapter on groups but in the Galois theory chapter. All these make for confusion for the student. I'd recommend Herstein's Topics in Algebra or Fraleigh's A First Course in Abstract Algebra instead for most students.
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