Algebraic Numbers and Algebraic Functions: 4 (Chapman Hall/CRC Mathematics Series) - Hardcover

Cohn, P.M.

Hardcover

ISBN 10: 0412361906 ISBN 13: 9780412361906

Publisher: Chapman and Hall/CRC, 1991

Synopsis

This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.

About this title

Review

"This elegant, lucid and highly readable exposition is strongly recommended to anyone with an interest in algebra." -Mathematika "The book is pleasantly written...it will certainly be useful to beginners in the field." -Zentralblatt fur Mathematik

Synopsis

This volume provides an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both, using E.Artin's approach, via valuations. A comprehensive account of valuations is included, as a powerful method of studying general fields, and a natural way of introducing Dedekind domains. Number theory is pursued as far as the unit theorem and the finiteness of the class number. The longest chapter in the book is on algebraic function theory; assuming, besides algebra, only a minimum of complex variable theory, it develops the subject from scratch, via the Riemann-Roch theorem to the Abel-Jacobi theorem which describes the divisor class group. An outline of Riemann surfaces is taken far enough to identify the topological significance of the genus, while occasional geometrical asides indicate the role of the genus for algebraic curve. Elliptic function fields and their automorphisms are explored in more detail, including Picard's theorem with an outline of the original proof. For fields of higher genus the finiteness of the automorphism group is established using Weierstrass points.

A brief final chapter looks at valuations of two-dimensional function fields, following Zariski, as a neat application of valuations and to illustrate the earlier theory. The point of view throughout the book is algebraic. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry, by an exposition of the central themes in the subject.

Edition details

• Publisher: Chapman and Hall/CRC
• Publication date: 1991
• Language: English
• ISBN 10: 0412361906
• ISBN 13: 9780412361906
• Binding: Hardcover
• Edition number: 1
• Number of pages: 192