The book under review is the unabridged English translation from the second edition of the French original published in 1997 (Zbl 0887.11003). Accordingly, and as it was already pointed out in the very thorough review of the first French edition, this text provides a sweeping introduction to all those mathematical topics, concepts, methods, techniques, and classical results that are necessary to understand Andrew Wiles’s theory culminating in the first complete proof of Fermat’s last theorem. Wiles presented his sensational proof in 1994, that is about 350 years after Fermat’s lapidary challenging formulation of his most famous, notorious number-theoretic “theorem” which remained, just as long, a haunting conjecture for generations of mathematicians. In the long meantime, an enormous number of mathematical developments, partly propelled by the various attempts of tackling the Fermat conjecture, had occured, and those finally helped lay the foundations for Wiles’s just as marvellous as intricate proof of Fermat’s last teorem.
The present book, grown out of a university course taught immediately after the appearance of Wiles’s proof, collects and expounds those multifarious and decisive developments in a very systematic, comprehensive and adapted manner, with a wealth of historical references and methodological matchings. Along the path to the gate of Wiles’s and his outriders’ (G. Frey, J.-P. Serre, K. Ribet, A. Weil, G. Shimura, Y. Taniyama, Y. Hellegouarch himself, and others) magic world of arithmetic and geometry, the author leads the reader into the realms of elliptic functions, Galois theory, elliptic curves, modular functions, and various recent conjectures related to Fermat’s last theorem, ranging from the Serre conjectures up to the famous “\(abc\)-conjecture” by J. Oesterlé and D. W. Masser (1985).
As for the more precise contents of the six chapters of the book, we refer again to the review of the French original (Zbl 0887.11003), as those have been left unaltered in the English translation under review. However, instead of repeating that part of the exhaustive first review of the book, it seems to be more appropriate to emphasize again the main features of this outstanding, utmost useful text.
First of all, this is not a research monograph addressed to specialists in the field. On the contrary, the text is accessible, without compromising the rigor of its mathematical exposition, to seasoned undergraduate students, at least so for the most part. On the other hand, the text offers an introduction to a wide range of classical and non-classical mathematical theories so that it can serve as the basis for various teaching courses, apart from such aiming at Fermat’s last theorem and ranging from topics in algebra, number theory, and representation theory up to complex analysis and geometry. Also, it touches upon many precious classical aspects that are no longer taught, and it sets the whole discussion in a fascinating, generally educating historical context, thereby travelling – metaphorically speaking – through the centuries of mathematical history. Finally, the book enraptures by the master’s pedagogical choices. He presents many non-classical proofs of classical results, using new ad hoc ideas, which makes the entire story really comprehensible for non-experts. As for the deeper structure theorems being beyond the scope of the book, the author provides the reader with detailed hints for further reading, and the numerous – often quite demanding – exercises effectively complement the material in many other directions, motivate the reader for deeper studies and lead him even closer to the forefront of current research in the field.
No doubt, it is a true blessing that the English translation of this unique book is now at hand for a much wider public.