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Lectures on Algebra, Volume I

S. S. Abhyankar
World Scientific
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Darren Glass
, on

When your intrepid MAA Reviews editor wrote me and asked me to review Shreeram Abhyankar's latest book Lectures on Algebra, he said that he thought I would find it particularly interesting and I believe he used the word "quirky" to describe the book. And despite all of the limitations of email as a form of communication, I could hear the irony in his voice from 600 miles away. The meaning of this became clear to me when my department administrator brought me the book — at 756 pages it wouldn't fit in my mailbox — which is volume one of a two-volume textbook surveying all of algebra.

The book consists of six "lectures" and the first lecture begins by discussing how one can complete the square to find the roots of the equation x2 – 3x – 18 = 0. By page twenty the author has defined the concepts of groups, fields, rings, ideals, modules, vector spaces, principal ideal domains, splitting fields, and several other topics that I would expect to be found towards the end of most books entitled "Lectures on Algebra." Something else happened by page twenty as well: I was hooked. It was at times frustrating, at times perplexing, and at times very charming, but this book was highly addictive and I stuck with it through all 756 pages, making Abhyankar join the ranks of J. K. Rowling and Tom Wolfe as the only authors that have captivated me for that size of a book.

Typically this would be the point in a review where I would list chapter titles and topics, but I'm not sure how useful that would be in the case of Abhyankar's book. For example, would you expect the lecture entitled "Quadratic Equations" to include the Fuundamental Theorem of Galois Theory or the lecture on "Curves and Surfaces" to include Zorn's Lemma and meromorphic series? (For what it is worth, however, the other lectures are entitled "Tangents and Polars", "Varieties and Models", "Projective Varieties", and "Pause and Refresh", and Abhyankar's fondness for algebraic geometry becomes clear throughout his exposition as well as his choice of topics).

It is worth pointing out that these topics do not get equal coverage in the book — section five of the fifth lecture takes 360 pages (The subsection on Projective Modules over Polynomial Rings alone accounts for 73 of these pages) while section two of the first lecture takes a third of a page and each of the first three lectures take up 30 pages. Did I mention that the book is quirky? If I still haven't captured the feel of this book, I should note that each lecture ends with a 'Concluding Note' which can be viewed as the moral of the lecture. The Concluding Note at the end of the first lecture is

Algebra means what you do not know, call it x. The experimental data will produce an equation in x. Solve it. The value of x so obtained will tell you what you wanted to know.

This passage also gives an example of the type of writing you are in store for if you open up Lectures on Algebra. Nobody can accuse Abhyankar of efficiency in his writing, and his exposition is a joy to read. When discussing the fact that the complex numbers are algebraically closed he writes "This used to be called the fundamental theorem of algebra, though it may be argued that it is neither fundamental nor a theorem of algebra." At a later point he discusses how he has "divided algebra into the High School Algebra of Polynomials and Power Series, the College Algebra of Rings and Ideals, and the University Algebra of Categories and Functors." When discussing the structure of the book, and its division into Lectures and Sections and Quests and Remarks and Observations, he writes that "A Problem is an Exercise which has not been completely worked out. By solving one of these, sometimes the student may get a mild satisfaction, sometimes a Ph.D. thesis, and sometimes fame. If there is some imprecision in the statement of a problem, a part of the exercise is to make it precise."

None of this will come as any surprise to people who have seen Abhyankar give a talk. I have had the pleasure of hearing him speak a number of times, and he rarely uses the chalkboard (let alone transparencies or fancy LaTeX presentations) instead just pontificating while occasionally writing something down. It always seems incredibly disorganized and inefficient yet at the end I feel as though I have learned something deep. So it came as no surprise that his book had a similar feel — oscillating between the extremely elementary and the highly technical, often taking digressions that seem completely random but turn out to contain enormous insight into the material.

To be blunt, I cannot imagine actually using this book as a textbook, or having it be your first introduction to many of the topics it contains. Abhyankar is extraordinarily inconsistent in the background he assumes — I noticed at least three times where he would use something on one page and then define it a dozen pages later — and sometimes his explanations are overly complicated or confusing. He also tends to shy away from concrete examples more than I would have needed when I first saw some of the material.

On the other hand, I cannot imagine that any mathematician would read even a few pages of this book without learning something new or seeing a new connection or gaining an insight into something that they thought they already knew. And between those insights and his writing, you will soon find yourself captivated by Abhyankar as well. I recommend that anyone with an interest in algebra find a copy this book and flip through it. And yes, you can hear the irony as I write that.

Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include Algebraic Geometry, Number Theory, and Cryptography. He can be reached at

  • Quadratic Equations (Rings)
  • Curves and Surfaces (Fields)
  • Tangents and Polars (Valuations)
  • Varieties and Models (Ideals)
  • Projective Varieties (Modules)
  • Pause and Refresh (Groups)