Science & Technology/ Mathematics Ash, Avner.  Fearless symmetry: exposing the hidden patterns of numbers, by Avner Ash and Robert Gross.  Princeton, 2006.  272p bibl index afp ISBN 0-691-12492-2, $24.95; ISBN 9780691124926, $24.95. Reviewed in 2007mar CHOICE. •  New from Princeton • 

Ash and Gross (both, Boston College) have prepared this expository book on algebraic number theory, one of the headiest and most arcane branches of modern mathematics. Their topics include groups, group representations, number systems, algebraic varieties, elliptic curves, and reciprocity laws. The surprise is the intended audience: the authors are writing for those who have studied only calculus. In fact, a reader of this book will certainly need more background than that, but Ash and Gross have done the best that probably can be done to make this very abstract material accessible at a relatively low level—by relying on examples and avoiding almost all proofs. Even so, the book makes dizzying reading as the abstract concepts pile up relentlessly. For the hardy reader, the reward is seeing described some of the ideas required by Andrew Wiles to prove Fermat’s Last Theorem.

Summing Up: Recommended. Graduate students through professionals. – M. Henle, Oberlin College

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