- 原书名：Class Field Theory (Universitext)
- 原出版社： Springer
1 A Brief Review
1 Number Fields
2 Completions of Number Fields
3 Some General Questions Motivating Class Field Theory
2 Dirichlet's Theorem on Primes in Arithmetic Progressions
1 Characters of Finite Abelian Groups
2 Dirichlet Characters
3 Dirichlet Series
4 Dirichlet's Theorem on Primes in Arithmetic Progressions
5 Dirichlet Density
3 Ray Class Groups
1 The Approximation Theorem and Infinite Primes
2 Ray Class Groups and the Universal Norm Index Inequality
3 The Main Theorems of Class Field Theory
4 The Idelic Theory
1 Places of a Number Field
2 A Little Topology
3 The Group of Ideles of a Number Field
4 Cohomology of Finite Cyclic Groups and the Herbrand Quotient
Also of interest is to describe how the prime ideals in the ring of integers of a global or local field decompose in its finite abelian extensions. In the case of the quadratic extensions of the field of rational numbers, such a description is obtained through the Law of Quadratic Reciprocity. There are also higher reciprocity laws of course, but all of these are subsumed by what is known as Artin Reciprocity, one of the most powerful results in class field theory.
I have always found class field theory to be a strikingly beautiful topic. As it developed, techniques from many branches of mathematics were adapted (or invented!) for use in class field theory. The interplay between ideas from number theory, algebra and analysis is pervasive in even the earliest work on the subject. And class field theory is still evolving. While it is prerequisite for most any kind of research in algebraic number theory, it also continues to engender active research. It is my hope that this book will serve as a gateway into the subject.
Class field theory has developed through the use of many techniques and points of view. I have endeavored to expose the reader to as many of the different tech-niques as possible. This means moving between ideal theoretic and idele theoretic approaches, with L-functions and the Tate cohomology groups thrown in for good measure. I have attempted to include some information about the history of the subject as well. The book progresses from material that is likely more naturally accessible to students, to material that is more challenging.
The global class field theory for number fields is presented in Chapters 2-6, which are intended to be read in sequence. For the most part they are not prerequisite for Chapter 7. (The exceptions to this are in Chapter 6: profinite groups and the theory of infinite Galois extensions in Section 6, and the notion of a ramified prime in an infinite extension from Section 7.) The local material is positioned last primar-ily because it is somewhat more challenging; for this reason, working through the earlier chapters first may be of benefit.
For students who have completed an introductory course on algebraic number theory, a one-term course on global class field theory might comprise Chapters 2-5 and sections 1--4 of Chapter 6. For more experienced students, some ofthe material in these chapters may be familiar, e.g., the sections on Dirichlet series and the The- orem on Primes in Arithmetic Progressions. In that case, the remainder of Chapter 6 may be included to produce a course still entirely on global class field theory. For somewhat more sophisticated students, Chapter 7 provides the option of including the local theory.
Facility with abstract algebra and (very) basic topology and complex analysis is assumed. Chapter I contains an outline of some of the prerequisite material on number fields and their completions. Nearly all of the results in Chapter 1 appear without proof, but details can be found in Frohlich and Taylor's Algebraic Number Theory, [FT], or (for the global fields) Marcus' Number Fields, [Ma].
The level of preparation in abstract algebra that is required increases slightly as one progresses through the book. However, I have included a little background material for certain topics that might not appear in a typical first-year course in abstract algebra. For example there are brief discussions on topological groups,infinite Galois theory, and projective limits. Finite Galois theory is heavily used throughout, and concepts such as modules, exact sequences, the Snake Lemma, etc.,play important roles in several places. A small amount of cohomology is introduced,but there is no need for previous experience with cohomology.
The source for the material on Dirichlet characters in Chapter 2 is Washington's Cyclotomic Fields, [Wa], while the material on Dirichlet series was adapted primar-ily from Serre's A Course in Arithmetic, [Se 1 ], and the book by Frohlich and Taylor,[Fr]. The section on Dirichlet density is derived mostly from Janusz' Algebraic Number Fields, [J], and Lang's Algebraic Number Theory, IL 1].
I first saw class fields interpreted in terms of Dirichlet density in Sinnott's lec-tures, [Si], which greatly influenced the organization of the material in Chapters 3 and 4.' (This point of view appears also in Marcus' Number Fields, [Ma].) Other sources that were particularly valuable in the writing of these two chapters were [J],[L1], and Cassels and Frohlich's Algebraic Number Theory, [CF].
The main source consulted in the preparation of Chapters 5 and 6 is [L1],although [J], [CF], [Si], Neukirch's Class FieM Theory, [NJ, and the lecture notes of Artin and Tate, [AT], also were very valuable throughout. For section 7 of Chapter 6,[Wa] is the primary source, and Lang's Cyclotomic Fields I and I1, [L3], was also consulted,
Other references that proved particularly useful in the preparation of the chap-ters on global class field theory include Gras' Class FieM Theory, [G], and Milne's lecture notes, [Mi].
The presentation of local class field theory in Chapter 7 relies mainly on the article by Hazewinkel, [Haz2]. Also very useful were Iwasawa's Local Class FieM Theory, III, Neukirch's book, [NJ, and the seminal article of Lubin and Tate, [LT].
A preliminary version of this book was used by a group of students and faculty at the University of Colorado, Boulder. I am indebted to them for their careful reading of the manuscript, and the many useful comments that resulted. My thanks espe-cially to David Grant, who led the group and kept detailed notes on these comments,and to the members: Suion Ih, Erika Frugoni, Vinod Radhakrishnan, Zachary Strider McGregor-Dorsey and Jonathan Kish.
Several incarnations of the manuscript for this book have been used for courses in class field theory that I have offered periodically. I am grateful to my class field theory students over the past few years, who have participated in these courses using early versions of the manuscript. Among those who have helped in spotting typo-graphical errors and other oddities are Eric Driver, Ahmed Matar, Chase Franks,Rachel Wallington, Michael McCamy and Shawn Elledge. Special thanks also to John Keri for advice on creating diagrams in LaTeX and to Linda Arneson for her excellent work in typing the first draft of the course outline, which grew into this book.
In completing this book, I am most fortunate to have worked with Mark Spencer,Frank Ganz and David Hartman at Springer, and to have had valuable input from the reviewers. My sincere thanks to them as well.