Great mathematicians are not always great expositors of mathematics; when they are, they are usually great in a certain way. They are not usually masters of mathematical prose. Instead, they are masters of mathematics. Having turned their attention to some elementary mathematics in order to explain it, they will think it through anew, understand it deeply, and therefore have something unique to say about it. Rather than writing the standard textbooks, they write the explanations that guide the textbook writers to the right way to present the mathematics. To quote the titles of a famous pair of articles in the Mathematical Intelligencer, they may not "say it better," but they certainly "think it gooder."
Emil Artin's expository work fits this description. Consider, for example, the three short books included in this volume. The Gamma Function is one of the great classics of mathematical exposition. In 38 pages, Artin tells us everything we need to know about the gamma function (over the reals), elegantly and efficiently. This booklet has been out of print for a long time, and its republication is reason for celebration.
Based on a set of lectures, Artin's Galois Theory has influenced a generation of books on the subject. It too is short and sweet: an 11-page summary of linear algebra followed by all of Galois theory in 25 pages. Artin's approach, very abstract and emphasizing linear algebra as the basic tool, has become the standard way of developing the subject at an advanced level. Rosen, in his introduction, explains that Artin did not like the traditional approach, which based Galois theory on the theory of symmetric functions and the primitive element theorem, and so re-cast the theory in this form.
The notes on the Theory of Algebraic Numbers are longer (127 pages), more standard, and less impressive, but they still give an efficient and useful account of the theory. Based on lectures given in the mid-1950s, they represent Artin's mature thoughts on how to present this material.
While exceptional students will enjoy the elegance of Artin's approach, others are likely to find it a little forbidding. Sometimes the efficiency sacrifices motivation completely. In fact, the book on Galois Theory includes a chapter by Arthur Milgram on the theory of polynomial equations and their solution by radicals, which Artin had not discussed at all. In most cases, there are few examples, so students trying to learn from these texts must be prepared to do quite a bit of work on their own. Of course, given Artin's insight, this work will usually be greatly rewarding — but it is not for the faint of heart.
In addition to these three short expository books, Exposition by Emil Artin includes a rather strange selection of papers. Some of these are truly expository: an account of the theory of braids that was originally published in American Scientist, an article on J. H. M. Wedderburn's work, and an article on complex function theory. Others are research papers deemed to be sufficiently elementary for inclusion here (three of these have been translated from the original German, a useful service). Finally, there is "A Proof of the Krein-Milman Theorem," originally a letter to Max Zorn, in which the reader finds himself in media res, without much clue as to which theorem is being proved, what the original publication was, or why a new proof is wanted. While there is much in these papers that is worth reading, for the most part they really don't qualify as "exposition." With few exceptions, readers who buy the book for the first half will prefer to ignore most of the second half.
Since the book is part of the "Sources" subseries of the AMS/LMS series on the History of Mathematics, one presumes that they are presented here as historical sources. What kind of history is in mind? Is it the history of mathematical expository writing in the 20th century? The history of number theory? The history of Artin's views of mathematics? Different parts of the book might serve for each of those purposes.
I was left wondering whether there weren't other sets of lecture notes that might have been included instead of the papers. A search using "WorldCat" revealed that the answer is yes, in fact: a book on algebraic topology, notes on algebraic geometry, a different set of notes on Galois theory, notes on rings with minimum condition, and even a Selected Topics in Modern Algebra . From a historian's point of view, it would have been useful to include a list of all of Artin's expository writing, and perhaps also to include one or two of the other books instead of the more technical articles.
In terms of production values, Exposition by Emil Artin is something of a mish-mash. Some pieces (The Gamma Function, some of the articles) are photographically reproduced from their original editions. In one case (page 8 of The Gamma Function, at least in my copy) the photograph missed a few characters at the the right end of the page; it is not enough to make the page unreadable, but it is disconcerting.
Other pieces have been retyped (and in some cases translated). Galois Theory, for example, was originally published as a photo-reproduced typescript that now looks quite old-fashioned, so it was reset in TeX. In the process, a few typos were (inevitably?) introduced; more annoyingly, a couple of typos in the original (two places in chapter 2, section I where a/b was used for a|b) were carefully preserved.
Photographed material has retained, of course, its original pagination, so that page 8 of The Gamma Function is also page 28 of Exposition by Emil Artin. When material was reset for this book, internal pagination was added (so, for example, section C of chapter two of Galois Theory starts on 15 of that book, which is page 79 of the overall book). The one exception is the paper on complex function theory, in which only the overall book pagination appears. When material was retyped or newly translated, the editor had the opportunity to add notes clarifying some of the harder passages. This was not possible, of course, in the pages that were photo-reproduced.
I am left with mixed feelings. While I am delighted to have this material in print and easily accessible, I wonder whether it might not have been better to reprint the three short books in three small volumes. I would love to have my real analysis students buy a copy of The Gamma Function, for example, but I hesitate to have them buy Exposition by Emil Artin simply to have access to those 40 pages.
Something similar is true about the papers: it is great to have translations of these three of Artin's papers, but it would have been much greater to have English translations of all of the papers that were written orginally in German. Since Artin's Collected Papers seems to be out of print, I'll be bold enough to suggest that a new edition including these and other translations would be very nice to have.
And, while I'm making suggestions to publishers, let me also suggest that a separate edition of Artin's pamphlet on the gamma function would make me very happy.
But despite the strange editorial choices and the issues of production, what matters in the end is the content. From that point of view, one must say that the three small books included here are worth the price of admission. They are mathematical exposition of the highest order: insightful, elegant, and rewarding.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.