This book deals with several aspects of what is now called "explicit number theory." 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.
From the reviews:
"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)
"The book deals with aspects of 'explicit number theory'. ... The central theme ... is the solution of Diophantine equations. ... It combines an interesting 'philosophy' of the subject with an encyclopedic grasp of detail. The extension of the author's reach via the contributed chapters is a good idea. Perhaps it is the start of a trend, as the subject grows more and more. ... It will undoubtedly be mined by instructors for their graduate courses, particularly for the purpose of including some recently-proved content." (R. C. Baker, Mathematical Reviews, Issue 2008 e)
"This is the second 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)