Graduate level textbooks II

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I would like to start with something at least a little bit shocking.

My first list will consists of books by two excellent authors who wrote many books each. These two authors are as different as one can imagine. I will say also few words about a third author, who worked nearly three hunderd years ago. The books mentioned in this post are not suggested for the first reading. I do not suggest them for reading cover to cover either. I will turn to more conventional books in the next post.


N. Bourbaki, Commutative algebra

N. Bourbaki, Lie groups and Lie algebras

N. Bourbaki, Elements of the History of Mathematics

N. Bourbaki, Théories spectrales

N. Bourbaki, Variétés différentielles et analytiques: fascicule de résultats

N. Bourbaki, Algèbre, Chapitre 10. Algèbre homologique


I do not suggest the more foundational books by Bourbaki; they are not suitable as textbooks at all. Actually, none of them is written as a textbook or intended to be one. The books listed above are written at a fairly advanced level. It is expected that the reader already has a motivation to study a particular area. These books have a perfect selection and organization of the material; proofs are condensed, but there is no handwaving and all the details are there. The book on manifolds contains no proofs; it is only a resume of the theory. The Chapter about homological algebra, probably, should be considered as outdated. But it hardly possible to start with the modern form of homological algebra; in any case, there is no textbook doing this.


Harold M. Edwards, Riemann's zeta function. For experts or to be experts only.

Harold M. Edwards, Advanced Calculus, A Differential Forms Approach. This is how one should teach calculus. I am not sure that there is any real need to study or teach calculus, but this is another topic.

Harold M. Edwards, Fermat's last theorem: a genetic introduction to algebraic number theory. Brilliant. But nobody planning to be an expert in algebraic number theory will have time to learn from this book, following the historical development of algebraic number theory.

Harold M. Edwards, Divisor Theory. This book is accessible and interesting, but very specialized.

Harold M. Edwards, Galois theory. If you know something about the Galois theory, it would be very instructive to take a look at what Galois really did.

Harold M. Edwards, Linear Algebra. This book is written at the undergraduate level. As always, Edwards takes a non-standard approach. It is good, but I do not suggest studying the linear algebra from it. Actually, one should not study the linear algebra as a separate subject at all. The reason is the fact that there is no such branch of mathematics, and never was such a branch.

Harold M. Edwards, Essays in Constructive Mathematics. Don’t be misled by the title; it is not about what people usually call “constructive mathematics”. It is an introduction to algebraic number theory and algebraic curves which stresses the explicit results (so that you can actually compute something) and the historical perspective.

Harold M. Edwards, Higher arithmetic: an algorithmic introduction to number theory. The title says it all.


The books by Harold M. Edwards are distinguished, first of all, by putting the material in the historical perspective. He follows the motto “Learn from the masters” and makes the works of discoverers accessible to the modern readers. The modern expositions are usually not only streamlined, but also watered down a lot, sometimes to the extent of eliminating all content. His later books also stress the algorithmic and computational aspects. This does not suits my tastes well, but it gives a new perspective, and when I read such good writer (I do not mean that this is easy), I can not only forgive, but also appreciate this.

I must admit that I did not read even a single chapter from the last two books, but they are on my reading list.

The history of a mathematical theory is its main and usually the only motivation (may be after an initial impetus from the outside). By this reason it makes a lot of sense to read not only 40 years old research papers (for a mathematician there is nothing unusual in this), but even 200 years old books. The problem is that they are written in a language hardly understandable now, and, in addition, they are usually written in Latin (the mathematical Latin is not very difficult but still is a serious obstruction). L. Euler is an exception. His books (and papers) are written in a way accessible to a modern reader. They are written in a style quite different from the modern one: Euler very often explains how he or his predecessors reached the presented results, and these explanations are an integral part of the text. They are not relegated to appendices at the ends of chapters or sections. Also, he wrote about results he wasn’t able to prove, explaining why there are compelling reasons to think that they are true.

Perhaps, every modern mathematician will be surprised by how far his textbooks in calculus go. Of course, they consist of several fairly extensive volumes. Still, this is the calculus of his time, and I doubt that many contemporary mathematicians will be able to master his more advanced topics (which include questions considered now as parts of algebraic geometry).

The main problem with Euler’s writings is the lack of English translations. It seems that all his books are translated into all main European languages except English. Still, something is translated. If you have time, his books are highly recommended. In fact, they can be even used in undergraduate teaching, if you are inclined to teach something meaningful and accessible and your undergraduate director will allow you to do this.


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