"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 MathematikVom Verlag:
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.
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Buchbeschreibung Chapman & Hall/CRC, 1991. Gebundene Ausgabe. First Edition. 192 Seiten Sehr gutes Exemplar. Sprache: Englisch Gewicht in Gramm: 391. Artikel-Nr. 38010