Synopses & Reviews
Review
From the reviews: "Number Theory, is poised to fill the gap as a core text in number theory ... . So, all in all, Henri Cohen's ... Number Theory are, to any mind, an amazing achievement. The coverage is thorough and generally all but encyclopedic, the exercises are good, some are excellent, some will keep even the best-prepared student busy for a long time, and the cultural level of the book ... is very high." (Michael Berg, MathDL, July, 2007) "The book under review deals with Diophantine analysis from a number-theoretic point of view. ... The clarity of the exposition is the one we expect from the author of two highly successful books on computational number theory ... and makes this volume a must-read for researchers in Diophantine analysis." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1119 (21), 2007) "Cohen (Université Bordeaux I, France), an instant classic, uniquely bridges the gap between old-fashioned, naive treatments and the many modern books available that develop the tools just mentioned ... . Summing Up: Recommended. ... Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (5), January, 2008) "This is the first volume of a highly impressive two-volume textbook on Diophantine analysis. ... Readers are presented with an almost overwhelming amount of material. This ... text book is bound to become an important reference for students and researchers alike." (C. Baxa, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)
Synopsis
The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second aspect is the global aspect: the use of number fields, and in particular of class groups and unit groups. The third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject, and embodies in a beautiful way the local and global aspects of Diophantine problems. In fact, these functions are defined through the local aspects of the problems, but their analytic behavior is intimately linked to the global aspects. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included 5 appendices on these techniques. These appendices were written by Henri Cohen, Yann Bugeaud, Maurice Mignotte, Sylvain Duquesne, and Samir Siksek, and contain material on the use of Galois representations, the superfermat equation, Mihailescu 's proof of Catalan 's Conjecture, and applications of linear forms in logarithms.
Synopsis
The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers.
Table of Contents
Preface.- Introduction to Diophantine Equations.- Abelian Groups, Lattices, and Finite Fields.-Basic Algebraic Number Theory.- p-adic Fields.-Quadratic Forms and Local-Global Principles.- Some Diophantine Equations.- Eilliptic Curves.- Diophantine Aspects of Elliptic Curves.- Bibliography.- Index.