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Learning Modern Algebra from Early Attempts to Prove Fermat's Last Theorem

Al Cuoco and Joseph J. Rotman
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Textbooks
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

This is an interesting, well-written book, in search of an appropriate course in which it could be used as a text.

From the title, one would think that it was intended primarily as a text for an introductory abstract algebra course, but using it that way would require a fairly radical overhaul of the traditional syllabus of such a course. This is intentional: the authors make clear in the Preface to the book that they believe that this traditional syllabus (namely number theory, followed by groups and then rings) to be not only “totally inadequate for future teachers of high school mathematics” but also “unsatisfying for other mathematics students” as well. They propose that abstract algebra should be taught in two semesters: number theory and rings in the first, groups and linear algebra in the second. Even for such a course, however, this book would likely not be appropriate for both semesters; it covers a lot of number theory and ring theory, but very little group theory and linear algebra. (More about the specific contents later.)

The primary intended audience of the book is future high school teachers. The authors take great pains to relate the material covered here to subjects that are taught in high school mathematics classes. And not just high school algebra classes: there is, for example, a fairly lengthy and quite detailed section on straightedge and compass constructions, including statements and (at least partial, and often full) proofs of many sophisticated results regarding impossible constructions. This section is one of the ones in the book captioned “Connections”; every chapter of the book except for the last one contains a section with this title, the goal of which is to relate the material taught to what is taught in high school. These Connections sections not only cover the substantive mathematics that is actually taught but also sometimes address pedagogical issues as well, including the use of complex numbers and the norm function to create problems that, in the authors’ words, “come out nice”, or comments addressed to the teacher that help put the material covered in perspective. So, one potential course that this book could be used for is one with a title like Mathematics for High School Teachers, but unfortunately many colleges do not offer courses along these lines.

As the title of the book indicates, it is organized along the lines of the history of Fermat’s Last Theorem (hereinafter FLT), although not every topic in the book is directly related to that major result. Most readers of this column will of course be familiar with this famous result and at least the outlines of its fascinating history: in the mid-17th century, Fermat wrote in the margin of his copy of Diophantus’ Arithmetica that he had discovered a “truly marvelous” proof of the fact the equation \(x^n+y^n=z^n\) had, for \(n > 2\), no solutions in positive integers. Mathematicians then spent about 350 years searching for a proof of this fact. A proof was not obtained until the mid-1990s by Andrew Wiles after seven years of solitary work; his proof uses a great deal of sophisticated mathematics not created until long after Fermat.

It is often stated that the modern subject of algebraic number theory was developed in response to attempts to prove FLT, and the authors suggest this in the preface when they state that “generalizations of prime numbers and unique factorization owe their initial study to attempts at proving Fermat’s Last Theorem.” See also page 218 of the text, where it is stated that “In his investigation of Fermat’s Last Theorem, Kummer invented ideal numbers in order to restore unique factorization.” However, as pointed out in the preface to Algebraic Number Theory and Fermat’s Last Theorem by Stewart and Tall, as well as Cox’s article “Introduction to Fermat’s Last Theorem” (American Mathematical Monthly, January 1994), the true impetus for Kummer’s work on ideal numbers was not FLT but rather his attempt to prove reciprocity laws. (By the way, early 1994 was a great year for Monthly articles on FLT; the March issue contains “A Marvelous Proof” by Fernando Gouvea — who, despite being the editor of this column, is in no way responsible for the inclusion of this reference!).

Although FLT may not have been the historical impetus for Kummer’s work on ideal numbers, there is an important connection between the two: if \(p\) is an odd prime, and \(\zeta\) denotes a primitive \(p\)-th root of unity, then \(x^p+y^p\) can be factored as the product of all terms \(x+\zeta^k y\) as \(k\) ranges from 0 to \(p-1\). This factorization takes place in the ring \(\mathbb{Z}[\zeta]\) obtained by adjoining \(\zeta\) to the ring of integers \(\mathbb{Z}\). If one assumes that this ring has unique factorization, one can prove FLT; in fact Lamé announced just such a proof in 1847. The assumption of unique factorization in this ring turns out to be false, however, as Kummer pointed out, and so this “proof” of FLT turns out to be invalid in general. So, FLT has connections to the concepts of ring and unique factorization; this fact plays a big role in this book, as will be seen a little later.

FLT is, in fact, related to a lot of other topics in mathematics as well, and even the case \(n=2\) of Fermat’s equation leads to some interesting mathematics. The text demonstrates this right from the beginning with an initial chapter on early number theory, which includes an extended discussion of the case \(n=2\) of Fermat’s equation and its connection to rational points on the circle. Geometric arguments are used to obtain a parameterization of such points. Also treated in this chapter is the case \(n = 4\) of FLT, and (in the Connections section) trigonometry and integration from the perspective of the parameterization of the unit circle. Obviously, there is much more number theory developed in this chapter than is typically covered in a beginning abstract algebra course.

Chapter 2 is a fairly standard introduction to mathematical induction, but is livened up by some interesting applications, such as differential equations and Fibonacci numbers. Chapter 3, entitled “Renaissance”, also begins with subjects not usually taught in an introductory abstract algebra course, namely the formulas for the roots of cubic and quartic polynomials, and proceeds to complex numbers, which of course pop up in these formulas. FLT does not loom terribly large in these chapters, but the Gaussian integers are defined, and a generalization of this ring, the ring of cyclotomic integers, is, as previously noted, related to an attempt to prove FLT.

The next chapter is entitled “Modular Arithmetic” but actually covers more than that: the idea of congruence modulo n is used to define the set of residue classes \(\mathbb{Z}_n\), which in turn leads in short order to the definition of commutative ring. Having thus begun the subject of abstract algebra, the authors continue this in some length through the next three chapters, which cover at least as much ring and field theory as the average student sees as an undergraduate: polynomial rings and their arithmetic, PIDs and UFDs (Euclidean domains don’t yet show up, but will appear later), extension fields, splitting fields, and finite fields. Along the way there are pleasant detours to interesting applications, such as Lagrange Interpolation and straightedge-compass constructions. (I prefer “straightedge” to the authors’ term “ruler” because no markings are allowed, but the authors do point this out explicitly, so the word “ruler” should cause no problems.)

The next chapter, a splendid one, puts all this abstract algebra to good use and really brings FLT into the picture. It returns to a theme mentioned earlier, namely the applicability of the ring of cyclotomic integers \(\mathbb{Z}[\zeta]\) (where \(\zeta\) is a primitive n-th root of unity) to FLT. The chapter first engages in a detailed look at the cases where \(n=3\) (Eisenstein integers) and \(n=4\) (Gaussian integers) and analyzes the structure of primes in these rings, and then applies that information to prove FLT for \(n=3\) (using unique factorization, which does exist here). Remarks are then made about the case of general integers n, and the chapter concludes with a Connections section applying this material to the question of counting the number of ways a positive integer can be written as the sum of two squares.

This chapter is rather technical and difficult, but the authors have made the details as palatable as possible for undergraduates. It is written so that even if some of the technical details are skipped, the underlying ideas are at least understandable. One typo I noticed that perhaps should be mentioned is that while the ring \(\mathbb{Z}[\zeta]\) is correctly defined on the very top of page 361, it was previously given a different and apparently incorrect definition in exercise 4.65 on page 168.

At this point, a lot of fairly sophisticated abstract algebra has been covered, but there is one fairly obvious omission that the reader may have noted: groups. In the next chapter, titled “Epilog”, they are finally introduced, in the context of the question of solvability of polynomial equations. The chapter begins with a brief historical discussion of Abel and Galois, discusses solvability of polynomials using field extensions (and the work on cubics and quartics done earlier), addresses the general idea of symmetry and then introduces groups and the basic terminology surrounding them (homomorphisms, normal subgroups, etc.) Galois groups are then used to explain why the general polynomial of degree 5 or greater is not solvable. The main result connecting solvable groups with polynomials that are solvable by radicals is stated but not proved; a reference is given to Rotman’s graduate text Advanced Modern Algebra.

The chapter ends with an expository section discussing Wiles’ work on FLT, which looks at groups in the context of rational points on elliptic curves. This material is surely beyond the capability of most undergraduates, but it’s nice to have an expository account of it, if only so that some of the better students can get an idea (even if it’s only a vague idea) of the ideas involved.

The book then ends with a set of Appendices covering both fairly routine issues (functions, equivalence relations, etc.) and a short but efficient look at vector spaces and linear transformations, giving the proof of the existence of a basis in the finite-dimensional case and also introducing dual spaces.

In writing this book, the authors have obviously kept the needs of the student-reader firmly in mind at all times. The writing style is not just clear, it is often conversational and humorous. Marginal notes appear frequently (as the authors point out, this seems particularly appropriate in a book on FLT), and there are frequent remarks captioned “How to Think about It”, which offer informal advice to the student about ways of thinking of the material. There is also more attention paid than usual in books of this nature to historical discussion, and also — a feature I have noted in other books by Rotman — there is a lot of attention paid to etymology, the origin of mathematical terms.

There are lots of exercises covering a wide range of difficulty, some with hints (but none with complete solutions) and there is also a pretty good 39-entry bibliography. One mild disappointment for me here was the fact that the bibliography consists entirely of books. I think it could have benefited from the inclusion of a few expository articles, not just because there are several good ones out there but also because I think having students look at journals occasionally is a valuable part of their training as mathematicians.

For the reasons expressed above, and because of the very interesting mathematical topics discussed in the text, I certainly have considerable admiration for this book as an example of mathematical exposition. I have some doubts, however, as noted earlier, as to whether this book would be suitable as a text for very many typical undergraduate courses. Perhaps it is just a matter of an old dog not being able to learn new tricks, but I remain a “groups first” person (and in some defense of this position, I point out that Rotman has written excellent undergraduate and graduate texts on abstract algebra from the “groups first” point of view). But even if I were to accept the premise that rings should come first, it seems strange and counter-intuitive to me to discuss groups after discussing more sophisticated topics like unique factorization. I wonder, therefore, whether this book will find much use in introductory abstract algebra courses.

What might be a very interesting use for this book would be as a text for a senior seminar or “topics” course for students who already have some prior exposure to abstract algebra. And, of course, whatever may be the applicability of this book as a text for an undergraduate course, it seems clear to me that it belongs in any good undergraduate library.

Mark Hunacek ( teaches mathematics at Iowa State University.


Some features of this book

A Note to Students

A Note to Instructors


1. Early Number Theory

2. Induction

3. Renaissance

4. Modular Arithmetic

5. Abstract Algebra

6. Arithmetic of Polynomials

7. Quotients, Fields, and Classical Problems

8. Cyclotomic Integers

9. Epilog