Introduction to Modern Number Theory [electronic resource] : Fundamental Problems, Ideas and Theories / by Yuri Ivanovic Manin, Alexei A. Panchishkin.
Contributor(s): Panchishkin, Alexei A [author.] | SpringerLink (Online service)Material type: TextSeries: Encyclopaedia of Mathematical Sciences: 49Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2005Edition: 2Description: XVI, 514 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783540276920Subject(s): Mathematics | Data encryption (Computer science) | Algebraic geometry | Mathematical logic | Number theory | Physics | Mathematics | Number Theory | Algebraic Geometry | Mathematical Logic and Foundations | Mathematical Methods in Physics | Data Encryption | Numerical and Computational PhysicsAdditional physical formats: Printed edition:: No titleDDC classification: 512.7 LOC classification: QA241-247.5Online resources: Click here to access online
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Problems and Tricks -- Number Theory -- Some Applications of Elementary Number Theory -- Ideas and Theories -- Induction and Recursion -- Arithmetic of algebraic numbers -- Arithmetic of algebraic varieties -- Zeta Functions and Modular Forms -- Fermat’s Last Theorem and Families of Modular Forms -- Analogies and Visions -- Introductory survey to part III: motivations and description -- Arakelov Geometry and Noncommutative Geometry (d’après C. Consani and M. Marcolli, [CM]).
"Introduction to Modern Number Theory" surveys from a unified point of view both the modern state and the trends of continuing development of various branches of number theory. Motivated by elementary problems, the central ideas of modern theories are exposed. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories. Moreover, the authors have added a part dedicated to arithmetical cohomology and noncommutative geometry, a report on point counts on varieties with many rational points, the recent polynomial time algorithm for primality testing, and some others subjects. From the reviews of the 2nd edition: "… For my part, I come to praise this fine volume. This book is a highly instructive read … the quality, knowledge, and expertise of the authors shines through. … The present volume is almost startlingly up-to-date ..." (A. van der Poorten, Gazette, Australian Math. Soc. 34 (1), 2007).