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Geometric Algebra

Emil Artin
Dover Publications
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Michael Berg
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It’s hard for me to fathom that it’s been close to forty years ago since my early graduate school days, when I first encountered Emil Artin; I wonder if he’s as big a player now as he was then. I’m inclined to think not, given that many of the books he was known for now find themselves in competition with a plethora of texts addressing current pedagogical needs, even at a beginning graduate school level. In my case, I first encountered Artin indirectly, through what is still my favorite graduate algebra text, B. L. van der Waerden’s Modern Algebra, which is famously based on the lectures of Artin and none other than Emmy Noether. Of course, Artin was himself influenced by the inimitable Noether, whom Alexandroff (another of her collaborators) christened der Noether. What a cast! Well, the professors I was taking higher algebra from, Ernst Straus and, later, Ian Morrison, had very classical tastes in algebra texts: the books referenced were not only the aforementioned classic, but (courtesy of Morrison) also Galois Theory by Artin himself. I recall that even in my naïveté I was enthralled by the style, clarity and concision of Artin’s writing: I remember seeing his proof of “linear independence of homomorphisms” for the first time and being duly impressed.

It was probably around that time that I learned that the most general version of number theoretic reciprocity was due to him, “Artin reciprocity,” and that his doctor’s thesis, written, I think, under Herglotz, dealt with quadratic function fields. All right, let’s look it up: Wikipedia confirms that Gustav Herglotz was his advisor, and it gives his thesis title as “On the Arithmetic of Quadratic Function Fields over Finite Fields.” I did know of his famous unfinished text of Class Field Theory, written with his pupil John Tate, and coming from their famous seminar (and featuring a very careful cohomological treatment of the two central inequalities in the subject), but didn’t have occasion to study this approach in detail: my de facto undergraduate advisor, V. S. Varadarajan, was giving a course on class field theory that was based on his own very detailed notes, flavored differently.

So Artin was certainly on my radar as I went on to study number theory for my doctorate, but the major players were really Erich Hecke and André Weil. It wasn’t until around three or four years after my PhD that I turned to another of Artin’s books (in order to broaden my horizons), namely, his terrific Algebraic Numbers and Algebraic Functions. Then, maybe five years or so later, I spent some time looking at his book with Hel Braun providing an Introduction to Algebraic Topology: I dare say that the discussion given there of the business of constructing the connecting homomorphisms in forming long exact sequences in (co)homology is arguably superior to others, if only because of, again, style, clarity and concision.

What does all this lead up to, then? Well, simply put: even in these days of pedagogical evolution, replete with shifting sands as far as curricula are concerned, it is still the case that getting mathematics under your belt by studying classical exposition with (bis) style, clarity and concision is highly desirable. You simply learn the stuff better.

These remarks certainly apply to the book under review: Artin’s Geometric Algebra. But the reader should be warned: Artin is going off the beaten track. In the Preface he states that while

[m]any parts of classical geometry have developed into great independent theories… [e.g.] linear algebra, topology, differential and algebraic geometry [,] it is frequently desirable to devise a course of geometric nature which is distinct from these great lines of thought and which can be presented to beginning graduate students or even to advanced undergraduates,

and hence we get Geometric Algebra. Despite the foregoing disclaimer, it has some very sexy stuff in it. After doing a good deal of linear algebra, group theory, and field theory in his first chapter, Artin presents “Affine and Projective Geometry” in Chapter II, “Symplectic and Orthogonal Geometry” in Chapter III, “The General Linear Group” in Chapter III, and finally “The Structure of Symplectic and Orthogonal Groups” in Chapter V. This last chapter even includes material on Clifford algebras and spinors. So there.

What more is there to say? Artin is always more than worth reading, even (or maybe especially) in these latter days, and the material he covers in this book clearly transcends many boundaries, with geometers, number theorists and even physicists called to play with the things Artin discusses so beautifully in these five chapters. I’m once again reminded of Abel’s pedagogical admonition that we should read “the masters, not their pupils.”

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.