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Algebraic Number Theory and Fermat's Last Theorem

Ian Stewart and David Tall
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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Back in 1847, when Fermat’s Last Theorem (asserting the non-existence of positive integer solutions to the equation \(x^n+y^n=z^n\) where \(n\) is a positive integer greater than 2) was still just a conjecture, the mathematician Gabriel Lamé announced that he had found a proof. Assuming that \(n\) is an odd prime (it is easy to see that FLT is true if it can be proved for such \(n\)), his proof involved assuming a solution, and then factoring the left hand side of the above equation, not in the ring of integers, but in the larger (cyclotomic) ring \(\mathbb{Z}[\zeta]\) obtained by adjoining a primitive \(n\)-th root of unity \(\zeta\) to the ring \(\mathbb Z\) of integers. In \(\mathbb{Z}[\zeta]\), the left hand side above factors as the product of all terms \(x+\zeta^iy\) as \(i\) ranges from \(0\) to \(n-1\). Lamé argued that it could be assumed that all the terms in this product were relatively prime, and, using the principle that the product of relatively prime terms is a prime power if and only if each term is, derived a contradiction.

Passing from \(\mathbb Z\) to \(\mathbb{Z}[\zeta]\) is a very clever idea, but, unfortunately, the proof is faulty. The argument that each term had to be a prime power would certainly be true in the ring \(\mathbb Z\) of integers, but that conclusion is based on the uniqueness of prime factorizations in \(\mathbb Z\) and, it turns out, factorization into primes is not necessarily unique in the various rings \(\mathbb{Z}[\zeta]\). Liouville pointed out the implicit assumption of unique factorization and also later announced that Kummer had shown, a few years before, that unique factorization can fail. Kummer himself subsequently came up with a proof of FLT for certain exponents \(n\); this proof used a concept he called “ideal numbers”, which restored a kind of unique factorization by resorting to elements from outside \(\mathbb{Z}[\zeta]\). These “ideal primes” eventually morphed into the concept that we now know of as a prime ideal.

The elements of \(\mathbb{Z}[\zeta]\) are examples of algebraic integers — i.e., a complex number that is a root of a nonconstant monic polynomial with integer coefficients. If we drop the requirement that the polynomial be monic, we get the notion of an algebraic number. By clearing denominators, we see that a complex number is an algebraic number if and only if it is a root of a monic (nonconstant) polynomial with rational coefficients. The relationship between algebraic numbers and algebraic integers is therefore roughly analogous to the relationship between rational numbers and ordinary integers, but the analogy is imperfect because unique factorization may fail. However, we can in some sense recover unique factorization in a ring of algebraic integers by passing from the elements of the ring to certain kinds of ideals. And thereby hangs a very interesting tale, namely the branch of mathematics known as algebraic number theory.

The above brief historical sketch illustrates that unique factorization (or the lack of it) is relevant to algebraic number theory and also to FLT. This book explores these connections. It can be viewed as an introduction to algebraic number theory, and also an introduction to Fermat’s Last Theorem. Along the way, other areas in which the study of algebraic numbers pays dividends in the study of ordinary integers are explored as well.

The book is divided into four parts. Part I, emphasizing algebra, introduces algebraic number theory from an ideal-theoretic perspective. Factorization theory, particularly as pertains to the ring of integers of an algebraic number field, is discussed, and quadratic and cyclotomic fields are singled out for detailed scrutiny. Applications of unique factorization to the solution of some particular Diophantine equations are also provided.

Part II of the book has a more geometric flavor. Lattices are introduced, and Minkowski’s theorem on volume is proved. This result is then applied to give proofs of the theorems characterizing those integers that can be written as the sum of two or four squares. A little later, the class group is defined and Minkowski’s theorem is also used to prove its finiteness.

Part III (“Number-Theoretic Applications”) begins with a chapter on computational methods in class number calculations, and then proceeds through a series of chapters (discussing such things as Kummer’s approach, elliptic curves, modular forms, etc.) to give an overview, almost a hundred pages long, on Wiles’s proof of FLT. Some proofs are completely omitted, of course; others are sketched, and some are given in reasonable detail.

Finally, part IV contains two appendices — one on quadratic residues, including both a statement and proof of the Law of Quadratic Reciprocity, and another stating and proving Dirichlet’s Units Theorem.

Readers familiar with the third edition of this text (published around fifteen years ago) will notice several changes. Some of these are purely cosmetic (a change in font in certain cases, for example) while others are more substantive: recent results about unique prime factorization in rings of integers of quadratic fields have been cited, and a chapter on elliptic functions in the previous edition has now been split into two chapters and enlarged. In addition, in the interim between editions, the Catalan conjecture (i.e., that 8 and 9 are the only consecutive powers of positive integers) has been solved by a method involving cyclotomic integers, and that is given a multi-page discussion here.

One thing that was not changed, but should have been, is a mistake in the statement of Mazur’s Theorem (page 232) characterizing the possibilities for the torsion subgroup of the group of rational points on an elliptic curve. In both the third edition and this one, the authors give an incomplete list by omitting \(\mathbb{Z}/12\mathbb{Z}\) as a possibility. (See, for example, Rational Points on Elliptic Curves by Silverman and Tate.)

Another change from the third edition concerns the back cover, which originally contained a blurb asserting that the book was “accessible to undergraduates.” No comparable statement appears on the back cover of the fourth edition, however, and that is probably a good thing: although this book is clearly written and is among the more reader-friendly of the various texts on the subject of algebraic numbers, most undergraduates (in the United States, at any rate) would likely, for several reasons, have trouble with this text.

For one thing, there is the matter of prerequisites. This book presupposes a basic understanding of some topics that many students just never see in their undergraduate courses, including field extensions, modules, symmetric polynomials, free abelian groups. These topics are discussed in the first chapter, but, as is often the case, the discussion is sufficiently rapid that somebody with no prior background in these areas would feel a little adrift. Later in the book, complex analysis is used.

In addition, most undergraduates would probably like to see more worked out examples than are presented here. The definition of “algebraic number”, for example, is not immediately followed by several examples of them, as is the case in, for example, Algebraic Number Theory by Jarvis. (Jarvis also mentions the classical result that \(\pi\) is not algebraic. That fact is mentioned in Stewart and Tall, but not until much later in the book, as part of a historical discussion.) So, as a text for one of those very rare undergraduate courses in algebraic number theory, I think Jarvis is probably a better bet, although Jarvis doesn’t talk about FLT.

This is not to say, however, that the book under review does not have much to recommend it. There is a nice emphasis on history, and, as noted above, the writing style is generally quite clear, given a reasonable degree of mathematical sophistication on the part of the reader. Moreover, the discussion of the proof of FLT strikes a very appealing balance — detailed enough to give a sense of the basic ideas of the proof, but not so detailed as to be incomprehensible. It is the discussion of FLT, I think, that sets this book apart from others — there are a number of other texts that introduce algebraic number theory, but I don’t know of any others that combine that material with the kind of detailed exposition of FLT that is found here.

To summarize and conclude: this is an interesting and attractive book. It would make an attractive text for an early graduate course on algebraic number theory, as well as a nice source of information for people interested in FLT, and especially its connections with algebraic numbers.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University. 

Algebraic Methods
Algebraic Background
Rings and Fields
Factorization of Polynomials
Field Extensions
Symmetric Polynomials
Free Abelian Groups


Algebraic Numbers
Algebraic Numbers
Conjugates and Discriminants
Algebraic Integers
Integral Bases
Norms and Traces
Rings of Integers


Quadratic and Cyclotomic Fields
Quadratic Fields
Cyclotomic Fields


Factorization into Irreducibles
Historical Background
Trivial Factorizations
Factorization into Irreducibles
Examples of Non-Unique Factorization into Irreducibles
Prime Factorization
Euclidean Domains
Euclidean Quadratic Fields
Consequences of Unique Factorization
The Ramanujan–Nagell Theorem


Historical Background
Prime Factorization of Ideals
The Norm of an Ideal
Nonunique Factorization in Cyclotomic Fields


Geometric Methods

The Quotient Torus


Minkowski's Theorem
Minkowski's Theorem
The Two-Squares Theorem
The Four-Squares Theorem

Geometric Representation of Algebraic Numbers
The Space Lst


Class-Group and Class-Number
The Class-Group
An Existence Theorem
Finiteness of the Class-Group
How to Make an Ideal Principal
Unique Factorization of Elements in an Extension Ring


Number-Theoretic Applications
Computational Methods

Factorization of a Rational Prime
Minkowski Constants
Some Class-Number Calculations
Table of Class-Numbers


Kummer's Special Case of Fermat's Last Theorem
Some History
Elementary Considerations
Kummer's Lemma
Kummer's Theorem
Regular Primes


The Path to the Final Breakthrough
The Wolfskehl Prize
Other Directions
Modular Functions and Elliptic Curves
The Taniyama–Shimura–Weil Conjecture
Frey's Elliptic Equation
The Amateur Who Became a Model Professional
Technical Hitch
Flash of Inspiration


Elliptic Curves
Review of Conics
Projective Space
Rational Conics and the Pythagorean Equation
Elliptic Curves
The Tangent/Secant Process
Group Structure on an Elliptic Curve
Applications to Diophantine Equations


Elliptic Functions
Trigonometry Meets Diophantus
Elliptic Functions
Legendre and Weierstrass
Modular Functions


Wiles's Strategy and Recent Developments
The Frey Elliptic Curve
The Taniyama–Shimura–Weil Conjecture
Sketch Proof of Fermat's Last Theorem
Recent Developments


Quadratic Residues

Quadratic Equations in Zm
The Units of Zm
Quadratic Residues


Dirichlet’s Units Theorem
Logarithmic Space
Embedding the Unit Group in Logarithmic Space
Dirichlet's Theorem