An invitation to the mathematics of Fermat-Wiles /

E.E. Kummer's approach to Fermat's Last Theorem, via cyclotomic fields, dominated research on the problem for nearly a century and a half, but Wiles's successful proof came along completely different lines. Wiles attacked Fermat's theories using elliptic curves, following a suggestion now generally tied to the name G. Frey, from 1985. It turns out that the author, and also B.A. Demjanenko, had already explored this connection as early as the 1960s. The problem's notoriety means that a large audience will be waiting for an accessible and faithful account, but the complexity of Wiles's methods makes such an account unlikely in the near term. The serious students must turn to Gary Cornell's Modular Forms and Fermat's Last Theorem (CH, Apr'98); Arithmetic Algebraic Geometry, ed. by B. Conrad and K. Rubin (2001); or the two recent books by Haruzo Hida, Geometric Modular Forms and Elliptic Curves (2000) and Modular Forms and Galois Cohomology (2000). Ignoring several shallow popularizations of dubious value, the dedicated mathematical tourist or inquisitive undergraduate may find that this book represents the unique, edifying, middlebrow initiation into the story. Fortunately, besides his bona fide credentials, Hellegouarch brings passion and a flair for exposition to the project. Highly recommended. Lower-division undergraduates and up. D. V. Feldman University of New Hampshire