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From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Development

W. Scharlau and H. Opolka
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
BLL Rating: 


[Reviewed by
Amy Shell-Gellasch
, on

I had, of course, heard of From Fermat to Minkowski, but it was still lingering on my “to read” list. So when asked to review this now 25 year old classic (is there an age requirement to call something a classic?) I jumped at the chance. As the title indicates, this text does double duty. It presents important gems in the theory of numbers as well as a short yet surprisingly comprehensive analysis of the historical development of the subject. Each of the ten lectures focus on an individual; tracing out their career and their impact on number theory: Fermat, Euler, Lagrange, Legendre, Gauss, Fourier, Dirichlet. The historical biographies are short but cover the important trends in each person’s life and career clearly and precisely, with special attention to trends in the development of number theory that are followed throughout the text. The last two lectures break this mold, covering number theory from Hermite to Minkowski in lecture nine, and providing an overview of reduction theory in lecture ten, reduction theory being one of the topical trends investigated throughout the text.

A short list of some of the other topics and trends discussed will give a feel for the coverage: the two-square theorem, Bernoulli numbers, Euler and the zeta function, Fermat’s (Pell’s) equation, quadratic reciprocity, continued fractions, and integral quadratic forms. The mathematical portions are a mix of straight results, results with brief explanations as to motivation, and proofs. Overall, the exposition is direct without being too terse or too verbose. The mathematics covered is in fact dense, despite the back cover’s claim that “this book can be read by advanced undergraduates… [or] anyone with an interest in number theory.” Anyone who is interested in number theory and has a good grounding in the subject will relish this volume.

The authors chose their content wisely. The items included show the versatility and beauty of number theory, as well as its breadth and intricacies. The other beauties included in this book are some historical gems that I was pleasantly surprised to find. In particular, there are quotes and excerpts from original sources and a nice array of photos and portraits. Special mention must be made of the following inclusions: a lengthy quote from the preface of Fourier’s Théorie Analytique de la Chaleur (his famous work on heat, 1822), images and a English translation of Gauss’s 1838 letter to Dirichlet, and a long excerpt from E. E. Kummer’s memorial of Dirichlet. These items are perfect additions to the more technical history provided.

The authors state that their text is meant, “…not as a systematic introduction to number theory but rather as a historically motivated invitation to the subject… to show how, in the historical development, the investigation of obvious or natural questions has led to more and more comprehensive and profound theories, how again and again, surprising connections between seemingly unrelated problems were discovered, and how the introduction of new methods and concepts led to the solution of hitherto unassailable questions.” I think this in fact describes From Fermat to Minkowski very well. So I also invite you to sample the large array of morsels they lay out.

Amy Shell-Gellasch is currently a Visiting Assistant Professor of Mathematics at Beloit College in Wisconsin. Her work is on the History of Mathematics and Its Uses in Teaching. Her most recent publication is the MAA Notes volume Hands On History which provides ways for teachers to make and use historical models in the mathematics classroom. She is very involved in the MAA and the HOM SIGMAA. She has organized numerous meetings and sessions to include the 2009 JMM MAA short course, Exploring the Great Books of Mathematics.




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