Introduction to Modern Number Theory

The book offered to the reader is a revised and expanded version of the book "Number Theory I. Introduction to Number Theory" by Yu. I. Manin and A. A. Panchishkin (Moscow: VINITI, 1989) and its English translation (Encyclopaedia of Mathematical... Sciences, v. 49, Springer-Verlag, 1995). The book consists of introductory chapters on various sections of number theory. All chapters are united by a common concept: to walk with the reader from visual examples of number-theoretic objects and problems, through general concepts and theories developed over a long time, to some of the newest achievements and visions of modern mathematics and sketches for further research. New sections written for this edition include a condensed presentation of Wiles' proof of Fermat's Last Theorem, a recently discovered polynomial algorithm for primality testing, a review of counting rational points on varieties, and other topics; the concluding part of the book is devoted to arithmetic cohomologies and noncommutative geometry.