Fundamentals of Advanced Mathematics 1

This volume is the first part of a projected trilogy that will discuss, in textbook form, a wide variety of topics in advanced mathematics, particularly those topics that find applications to systems theory, physics and engineering.

Volume 1, the book now under review, focuses on algebra. It is divided into three chapters: the first is on category theory, and includes a ten-page discussion (under “category of sets”) of the basics of set theory, including cardinal number arithmetic, Zorn’s Lemma and the Axiom of Choice, as well as a brief look at well-ordering and ordinal numbers. The second chapter is on algebraic structures. It starts with monoids, quickly gets to groups, and proceeds from there to rings (which are assumed to have an identity) and modules, including extensive discussion of the noncommutative theory. The final chapter deals with algebras and modules, including a look at commutative algebra (prime and maximal ideals, primary decomposition, Krull dimension, affine varieties, etc.), modules over noncommutative rings (including noncommutative principal ideal domains) and homological algebra (including projective and injective modules, homology and cohomology).

The exposition here is very much in the French style — in particular, it gives whole new meaning to the word “concise”. (I can’t resist saying, as did Sherlock Holmes of Moriarty’s treatise on the binomial theorem, that it has a “European vogue”.) Most of the material here is found in standard graduate algebra textbooks such as those by Dummit and Foote, Rotman or Hungerford, but things are done much more quickly here — the fact that all the material summarized above (as well as other topics) is done in just about 250 pages of text should provide ample testament to that fact. As a specific example of the speed with which these topics are covered, note that groups are first defined on page 40 and by page 52 the author has covered Lagrange’s theorem, cyclic subgroups, normal subgroups and quotient groups, solvable and nilpotent groups, and group actions, including a proof of Burnside’s orbit-counting theorem.

The author’s terminology is often nonstandard, at least by American standards. He uses, for example, the phrase “entire ring” to mean a ring with no zero divisors, and says that some other authors use the term “integral domain” for this. However, as he also points out, many other authors assume integral domains are commutative, but he does not. I had also, prior to looking at this book, never heard of a “semifir”, defined here to be a ring in which every finitely generated left ideal is free, with a unique rank.

Elsevier’s webpage for the book describes the “readership” of this text as including undergraduate students in mathematics, but this really can’t be taken seriously. No undergraduate student of my acquaintance would likely derive anything from this text other than feelings of confusion and inadequacy. Indeed, the book itself is more realistic: the author describes the target audience for the three volumes as students who have “mastered the most important parts of a Mathematics degree or the mathematical content taught in most advanced engineering courses.” He lists, as textbooks that are “amply sufficient” for prerequisites, Godement’s algebra text, Dieudonné’s Foundations of Modern Analysis, and also suggests a prior understanding of “a few notions of measure theory and integration”.

Of course, when one reads about a French multi-volume treatise on a wide range of mathematical topics, the name Bourbaki springs immediately to mind. The influence of Bourbaki on this text is apparent (and Bourbaki is frequently quoted), but there are some substantial differences as well. For one thing, this book is much shorter than Bourbaki, and the author here does not feel the need to prove everything. He omits, for example, the proof of the Artin-Weddeburn theorem on semisimple Artinian rings, which is something that I can’t imagine Bourbaki doing.

What can we expect in future installments of this trilogy? Volume 2, the publication of which is, according to Elsevier’s webpage, set for late January 2018, will cover more algebra (field extensions, Galois theory, differential Galois theory, sheaves), as well as some topology and functional analysis (topology and topological vector spaces, function spaces, differentiation in Banach spaces, distributions). The contents (and publication date) of Volume 3 are, at this point, unknown to me; the text does not mention this volume, and I was unable to find any reference to it on the Elsevier webpage.

To summarize and conclude: given the amount of material discussed here and the concise way in which it is presented, graduate students and faculty members will likely find this book a compact and useful reference. As a text for graduate students, however, it does not, in my opinion, represent a pedagogical advance on the standard graduate algebra books that are already available, all of which are much more accessible to a student audience.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.