<p>Modern number theory, according to Hecke, dates from Gauss's quadratic reciprocity law. The various extensions of this law and the generalizations of the domains of study for number theory have led to a rich network of ideas, which has had effects throughout mathematics, in particular in algebra. This volume of the Encyclopaedia presents the main structures and results of algebraic number theory with emphasis on algebraic number fields and class field theory. Koch has written for the non-specialist. He assumes that the reader has a general understanding of modern algebra and elementary number theory. Mostly only the general properties of algebraic number fields and related structures are included. Special results appear only as examples which illustrate general features of the theory. A part of algebraic number theory serves as a basic science for other parts of mathematics, such as arithmetic algebraic geometry and the theory of modular forms. For this reason, the chapters on basic number theory, class field theory and Galois cohomology contain more detail than the others. This book is suitable for graduate students and research mathematicians who wish to become acquainted with the main ideas and methods of algebraic number theory.</p>

Number Theory II: Algebraic Number Theory (Encyclopaedia of Mathematical Sciences)

Science & Math

A.N. Parshin

Mathematics

Springer, Verlag

<p>This volume of the Encyclopaedia contains two contributions on closely related subjects: the theory of linear algebraic groups and invariant theory. The first part is written by T.A. Springer, a well-known expert in the first mentioned field. He presents a comprehensive survey, which contains numerous sketched proofs and he discusses the particular features of algebraic groups over special fields (finite, local, and global). The authors of part two, E.B. Vinberg and V.L. Popov, are among the most active researchers in invariant theory. The last 20 years have been a period of vigorous development in this field due to the influence of modern methods from algebraic geometry. The book will be very useful as a reference and research guide to graduate students and researchers in mathematics and theoretical physics.</p>

Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory (Encyclopaedia of Mathematical Sciences)

Algebra VII: Combinatorial Group Theory Applications to Geometry (Encyclopaedia of Mathematical Sciences)

This volume of the EMS consists of two parts. The first entitled Combinatorial Group Theory and Fundamental Groups, written by Collins and Zieschang, provides a readable and comprehensive description of that part of group theory which has its roots in topology in the theory of the fundamental group and the theory of discrete groups of transformations. Throughout the emphasis is on the rich interplay between the algebra and the topology and geometry. The second part by Grigorchuk and Kurchanov is a survey of recent work on groups relating to topological manifolds, dealing with equations in groups, particularly in surface groups and free groups, a study in terms of groups of Heegaard decompositions and algorithmic aspects of the Poincaré conjecture, as well as the notion of the growth of groups. The authors have included a list of open problems, some of which have not been considered previously. Both parts contain numerous examples, outlines of proofs and full references to the literature. The book will be very useful as a reference and guide to researchers and graduate students in algebra and topology.

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Books by A.N. Parshin

See the best price to sell, buy, or rent books by A.N. Parshin. BookScouter helps to compare book prices from 25+ online bookstores and 30+ buyback vendors with a single search.

Number Theory II: Algebraic Number Theory (Encyclopaedia of Mathematical Sciences)

Algebraic Geometry IV: Linear Algebraic Groups Invariant Theory (Encyclopaedia of Mathematical Sciences)

Algebra VII: Combinatorial Group Theory Applications to Geometry (Encyclopaedia of Mathematical Sciences)