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A Survey of Modern Algebra

A K Peters
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Garrett Birkhoff and Saunders Mac Lane published their Survey of Modern Algebra in 1941. The book was written because the authors could find no adequate text to use with their students at Harvard. Both had learned "Modern Algebra" from van der Waerden's famous text (originally published in 1930). Birkhoff had started teaching a course on the subject to Harvard undergraduates in the mid-thirties, and Mac Lane took over the course when he arrived at Harvard (from Göttingen) in 1938. One has to admire the insight of the authors in judging that the "modern" approach was the way of the future and their willingness to write a book that would be accessible to undergraduates. It was largely through this book, supplemented later by I. N. Herstein's Topics in Algebra, that the new algebraic ideas were incorporated into American mathematics education.

In contrast to many other authors, Birkhoff and Mac Lane did not simply follow van der Waerden's table of contents, opting instead for a presentation that they felt would make the book more accessible. The first chapter discusses the integers as an example of a commutative ring, developing from the start a dialectic between theory and example. The integers modulo n also appear in this chapter. Then come the rational numbers and fields, then a fairly classical chapter on polynomials which includes a proof that partial fraction decompositions are always possible. Groups only show up in chapter 6. This is a very plausible approach to introducing students to algebra, but one followed today by few instructors (and fewer textbooks).

Some topics treated by Birkhoff and Mac Lane today seem out of place. Chapters 4 and 5 develop the algebra of real and complex numbers, including an (optional) section on Dedekind cuts and one (non-optional) giving a proof of the Fundamental Theorem of Algebra. Both would today appear, if they appear at all, in analysis courses. Chapters 7, 8, and 10 constitute a short course on linear algebra, which was not yet a part of the undergraduate curriculum. Chapter 12 is an introduction to set theory and transfinite arithmetic, which nowadays are usually treated either in a "transition to abstract mathematics" course or in advanced calculus/real analysis. Lattices and boolean algebras, the subject of chapter 11, never became a fixture of undergraduate education.

Some aspects of the book clearly were very influential. The prominence given to factorization theory in integral domains, for example, established that topic as a standard part of undergraduate abstract algebra. The authors' choice of topics in group theory still forms the basic core of the subject in most books, and their decision to de-emphasize the quotient construction has also seemed wise to many. The unfortunate notation Zn has become standard, to the frustration of those of us who feel it should denote the n-adic integers and not Z/nZ.

Other aspects were less successful. I think Birkhoff and Mac Lane were absolutely correct to include a chapter on the classical matrix groups, but their lead was largely not followed. Their notation Ln(K) for what is today usually denoted GLn(K) also didn't stick.

Given all this, adopting this as the main textbook for an undergraduate abstract algebra course would today be an eccentric move. Nevertheless, it is still a book well worth reading. 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. 

A famous mathematician once remarked to me that everyone he knew who had worked through A Survey of Modern Algebra had come to love the subject. That may overstate things, since my friend probably knows more mathematicians than students who got fed up and left. But the authors' delight in what was then a new subject shines through their writing, and their willingness to be informal when necessary was a smart move. 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. Birkhoff and Mac Lane also want to teach their students to prove things, of course. But they want to teach them algebra even more.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He learned abstract algebra at the University of São Paulo from César Polcino, who he now realizes was strongly influenced by Birkhoff and Mac Lane. 

Date Received: 
Monday, August 16, 2010
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Garrett Birkhoff and Saunders Mac Lane
AKP Classics
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Fernando Q. Gouvêa
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Monday, July 18, 2011