标题：分圆域
图书信息
出版社: 世界图书出版公司; 第2版 (2009年6月1日)
外文书名: Cyclotomic Fields
平装: 433页
正文语种: 英语
开本: 24
ISBN: 9787510004773, 7510004772
条形码: 9787510004773
尺寸: 22.2 x 14.8 x 1.8 cm
重量: 558 g
作者简介
作者：(美国)朗(Lang.S.)
内容简介
《分圆域(第2版)(英文版)》讲述了:Kummer's work on cyclotomic fields paved the way for the development ofalgebraic number theory in general by Dedekind, Weber, Hensel, Hilbert,Takagi, Artin and others. However, the success of this general theory hastended to obscure special facts proved by Kummer about cyclotomic fieldswhich lie deeper than the general theory. For a long period in the 20th centurythis aspect of Kummer's work seems to have been largely forgotten, exceptfor a few papers, among which are those by Pollaczek [Po], Artin-Hasse[A-H] and Vandiver .   In the mid 1950's, the theory of cyclotomic fields was taken up again byIwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analoguesfor number fields of the constant field extensions of algebraic geometry, andwrote a great sequence of papers investigating towers of cyclotomic fields,and more generally, Gaiois extensions of number fields whose Galois groupis isomorphic to the additive group ofp-adic integers. Leopoldt concentratedon a fixed cyclotomic field, and established various p-adic analogues of theclassical complex analytic
目录
Notation
Introduction
CHAPTER 1 Character Sums
1.Character Sums over Finite Fields
2.Stickelberger's Theorem
3.Relations in the Ideal Classes
4.Jacobi Sumsas Hecke Characters
5.Gauss Sums over Extension Fields
6.Application to the Fermat Curve
CHAPTER 2 Stickelberger Ideals and Bernoulli Distribution
1.The Index of the First Stickelberger Ideal
2.Bernoulli Numbers
3.Integral Stickelberger Ideals
4.General Comments on Indices
5.The Index for k Even
6.The Index for k Odd
7.Twistings and Stickelberger Ideals
8.Stickelberger Elements as Distributions 
9.Universal Distributions
10. The Davenport-Hasse Distribution
Appendix. Distributions
CHAPTER 3 Complex Analytic Class Number Formulas
1.Gauss Sums on Z/raZ
2.Primitive L-series
3.Decomposition of L-series
4.The (+ I)-eigenspaces
5.Cyclotomic Units
6.The Dedekind Determinant
7.Bounds for Class Numbers
CHAPTER 4
The p-adic L-function
1. Measures and Power Series
2. Operations on Measures and Power Series
3. The Mellin Transform and p-adic L-function
Appendix, The p-adic Logarithm
4. The p-adic Regulator
5. The Formal Leopoldt Transform
6, The p-adic Leopoldt Transform
CHAPTER 5
Iwasawa Theory and Ideal Class Groups
1. The lwasawa Algebra
2. Weierstrass Preparation Theorem
3. Modules over Zp[[X]]
4. extensions and Ideal Class Groups
5. The Maximal p-abelian p-ramified Extension
6. The Galois Group as Module over the lwasawa Algebra
CHAPTER 6
Kummer Theory over Cyclotomic Z,-extensions
1. The Cyclotomic extension
2. The Maximal p-abelian p-ramified Extension of the Cyclotomic extension
3. Cyclotomic Units as a Universal Distribution
4. The lwasawa-Leopoldt Theorem and the Kummer-Vandiver Conjecture
CHAPTER 7
Iwasawa Theory of Local Units
1. The Kummer-Takagi Exponents
2. Projective Limit of the Unit Groups
3. A Basis for U(X) over A
4. The Coates-Wiles Homomorphism
5. The Closure of the Cyclotomic Units
CHAPTER 8
Lubin-Tate Theory
1. Lubin-Tate Groups
2. Formal p-adic Multiplication
3. Changing the Prime
4. The Reciprocity Law
5. The Kummer Pairing
6. The Logarithm
7. Application of the Logarithm to the Local Symbol
CHAPTER 9
Explicit Reciprocity Laws
1. Statement of the Reciprocity Laws
2. The Logarithmic Derivative
3. A Local Pairing with the Logarithmic Derivative
4. The Main Lemma for Highly Divisible x and 
5. The Main Theorem for the Symbol
6. The Main Theorem for Divisible x and a, ffi unit
7. End of the Proof of the Main Theorems
CHAPTER 10
Measures and Iwasawa Power Series
I. Iwasawa Invariants for Measures
2. Application to the Bernoulli Distributions
3. Class Numbers as Products of Bernoulli Numbers Appendix by L. Washington: Probabilities
4. Divisibility by ! Prime to p: Washington's Theorem
CHAPTER 11
The Ferrero-Washington Theorems
I. Basic Lemma and Applications
2. Equidistribution and Normal Families
3. An Approximation Lemma
4. Proof of the Basic Lemma
CHAPTER 12
Measures in the Composite Case
1. Measures and Power Series in the Composite Case
2. The Associated Analytic Function on the Formal Multiplicativc Group
3. Computation of in the Composite Case
CHAPTER 13
Divisibility of Ideal Class Numbers
I. lwasawa lnvariants in extensions
2. CM Fields, Real Subfields, and Rank Inequalities
3. The/-primary Part in an Extension of Degree Prime to 
4. A Relation between Certain lnvariants in a Cyclic Extension
5. Examples of lwasawa
6. A Lemma of Kummer
CHAPTER 14
p-adic Preliminaries
1. The p-adic Gamma Function
2. The Artin-Hasse Power Series
3. Analytic Representation of Roots of Unity Appendix: Barsky's Existence Proof for the p-adic Gamma Function
CHAPTER 15
The Gamma Function and Gauss Sums
1. The Basic Spaces
2. The Frobenius Endomorphism
3. The Dwork Trace Formula-and Gauss Sums
4. Eigenvalues of the Frobenius Endomorphism and the p-adic Gamma Function
5. p-adic Banach Spaces
CHAPTER 16
Gauss Sums and the Artin-Schreier Curve
I. Power Series with Growth Conditions
2. The Artin-Schreier Equation
3. Washnitzer-Monsky Cohomology
4. The Frobenius Endomorphism
CHAPTER 17
Gauss Sums as Distributions
1. The Universal Distribution
2. The Gauss Sums as Universal Distributions
3. The L-function at s = 0
4. The p-adic Partial Zeta Function
APPENDIX BY KARL RUBIN
The Main Conjecture Introduction
1. Setting and Notation
2. Properties of Kolyvagin's "Euler System"
3. An Application of the Chebotarev Theorem
4. Example: The Ideal Class Group of
5. The Main Conjecture
6. Tools from lwasawa Theory
7. Proof of Theorem 5.1
8. Other Formulations and Consequences of the Main Conjecture
Bibliography
Index