Valued fields被重视的领域
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作者： Antonio J. Engler著
出 版 社： 北京燕山出版社
出版时间： 2005-11-1字数：版次： 1页数： 205印刷时间： 2005/11/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540242215包装： 精装编辑推荐
From the reviews:
"The book starts with the basic notion of absolute values followed by a comprehensive introduction to the theory of Krull valuations of arbitrary rank leading eventually to some deep results of recent research. … A useful feature of the book are its two appendices dealing with classification of V-topologies and ultraproducts of valued fields. The concise style and choice of material makes this book a wonderful reading. It is a unique, original exposition full of valuable insights." (Sudesh Kaur Khanduja, Zentralblatt MATH, Vol. 1128 (6), 2008)
内容简介
Absolute values and their completions -like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization.
In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge acquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -for instance to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values alone.
目录
Introduction
1 Absolute Values
1.1 Absolute Values ?Completions
1.2 Archimedean Complete Fields
1.3 Non-Archimedean Complete Fields
2 Valuations
2.1 Ordered Abelian Groups ?Valuations
2.2 Constructions of Valuations
2.2.1 Rational Function Fields
2.2.2 Ordered Fields
2.2.3 Rigid Elements
2.3 Dependent Valuations ?Induced Topology
2.4 Approximation ?Completion
2.5 Exercises
3 Extension of Valuations
3.1 Chevalley's Extension Theorem
3.2 Algebraic Extensions
3.3 The Fundamental Inequality
3.4 Transcendental Extensions
3.5 Exercises
4 Henselian Fields
4.1 Henselian Fields
4.2 p-Henselian Fields
4.3 Ordered Henselian Fields
4.4 The Canonical Henselian Valuation
4.5 Exercises
5 Structure Theory
5.1 Infinite Galois Groups
5.2 Unramified Extensions ?First Exact Sequence
5.3 Ramified Extensions ?Second Exact Sequence
5.4 Galois Characterization of Henselian Fields
5.5 Exercises
6 Applications of Valuation Theory
6.1 Artin's Conjecture
6.2 p-Adically Closed Fields
6.3 A Local-Global Principle for Quadratic Forms
A Ultraproducts of Valued Fields
B Classification of V-Topologies
References
Standard Notations
Index
