Original Russian version of the first edition was published by VINITI, Moscow in 1990
2nd ed. 2005, XVI, 514 p.
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With the second edition widely praised for its authoritative yet approachable style, in this corrected second printing you can be sure this excellent work is even more contemporaneous.
It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory.
Illuminated by elementary problems, the central ideas of modern theories are laid bare. 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.
What’s more, the authors have added a segment dedicated to arithmetical cohomology and noncommutative geometry as well as a report on point counts on varieties with many rational points.
Also covered is the recent polynomial time algorithm for primality testing, among other subjects.
From the reviews of the 2nd edition: "… For my part, I come to praise this fine volu
Content Level »Research
Keywords »Arithmetic - Arithmetic der algebraischen Zahlen - Elementare Zahlentheorie - Elementary number theory - Langlands program - Langlands-Programm - algebraic varieties - arithmetic of algebraic numbers - commutative property - diophantine equations - diophantische Gleichungen - elliptic curves - elliptische Kurven - number theory - public