【什么是分圆域-图】百科知识点

来源:学大教育    日期:2017-11-23 13:33:57

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【什么是分圆域-图】百科知识点

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

《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

NotationIntroductionCHAPTER 1 Character Sums1.Character Sums over Finite Fields2.Stickelberger's Theorem3.Relations in the Ideal Classes4.Jacobi Sumsas Hecke Characters5.Gauss Sums over Extension Fields6.Application to the Fermat CurveCHAPTER 2 Stickelberger Ideals and Bernoulli Distribution1.The Index of the First Stickelberger Ideal2.Bernoulli Numbers3.Integral Stickelberger Ideals4.General Comments on Indices5.The Index for k Even6.The Index for k Odd7.Twistings and Stickelberger Ideals8.Stickelberger Elements as Distributions9.Universal Distributions10. The Davenport-Hasse DistributionAppendix. DistributionsCHAPTER 3 Complex Analytic Class Number Formulas1.Gauss Sums on Z/raZ2.Primitive L-series3.Decomposition of L-series4.The (+ I)-eigenspaces5.Cyclotomic Units6.The Dedekind Determinant7.Bounds for Class NumbersCHAPTER 4The p-adic L-function1. Measures and Power Series2. Operations on Measures and Power Series3. The Mellin Transform and p-adic L-functionAppendix, The p-adic Logarithm4. The p-adic Regulator5. The Formal Leopoldt Transform6, The p-adic Leopoldt TransformCHAPTER 5Iwasawa Theory and Ideal Class Groups1. The lwasawa Algebra2. Weierstrass Preparation Theorem3. Modules over Zp[[X]]4. extensions and Ideal Class Groups5. The Maximal p-abelian p-ramified Extension6. The Galois Group as Module over the lwasawa AlgebraCHAPTER 6Kummer Theory over Cyclotomic Z,-extensions1. The Cyclotomic extension2. The Maximal p-abelian p-ramified Extension of the Cyclotomic extension3. Cyclotomic Units as a Universal Distribution4. The lwasawa-Leopoldt Theorem and the Kummer-Vandiver ConjectureCHAPTER 7Iwasawa Theory of Local Units1. The Kummer-Takagi Exponents2. Projective Limit of the Unit Groups3. A Basis for U(X) over A4. The Coates-Wiles Homomorphism5. The Closure of the Cyclotomic UnitsCHAPTER 8Lubin-Tate Theory1. Lubin-Tate Groups2. Formal p-adic Multiplication3. Changing the Prime4. The Reciprocity Law5. The Kummer Pairing6. The Logarithm7. Application of the Logarithm to the Local SymbolCHAPTER 9Explicit Reciprocity Laws1. Statement of the Reciprocity Laws2. The Logarithmic Derivative3. A Local Pairing with the Logarithmic Derivative4. The Main Lemma for Highly Divisible x and5. The Main Theorem for the Symbol6. The Main Theorem for Divisible x and a, ffi unit7. End of the Proof of the Main TheoremsCHAPTER 10Measures and Iwasawa Power SeriesI. Iwasawa Invariants for Measures2. Application to the Bernoulli Distributions3. Class Numbers as Products of Bernoulli Numbers Appendix by L. Washington: Probabilities4. Divisibility by ! Prime to p: Washington's TheoremCHAPTER 11The Ferrero-Washington TheoremsI. Basic Lemma and Applications2. Equidistribution and Normal Families3. An Approximation Lemma4. Proof of the Basic LemmaCHAPTER 12Measures in the Composite Case1. Measures and Power Series in the Composite Case2. The Associated Analytic Function on the Formal Multiplicativc Group3. Computation of in the Composite CaseCHAPTER 13Divisibility of Ideal Class NumbersI. lwasawa lnvariants in extensions2. CM Fields, Real Subfields, and Rank Inequalities3. The/-primary Part in an Extension of Degree Prime to4. A Relation between Certain lnvariants in a Cyclic Extension5. Examples of lwasawa6. A Lemma of KummerCHAPTER 14p-adic Preliminaries1. The p-adic Gamma Function2. The Artin-Hasse Power Series3. Analytic Representation of Roots of Unity Appendix: Barsky's Existence Proof for the p-adic Gamma FunctionCHAPTER 15The Gamma Function and Gauss Sums1. The Basic Spaces2. The Frobenius Endomorphism3. The Dwork Trace Formula-and Gauss Sums4. Eigenvalues of the Frobenius Endomorphism and the p-adic Gamma Function5. p-adic Banach SpacesCHAPTER 16Gauss Sums and the Artin-Schreier CurveI. Power Series with Growth Conditions2. The Artin-Schreier Equation3. Washnitzer-Monsky Cohomology4. The Frobenius EndomorphismCHAPTER 17Gauss Sums as Distributions1. The Universal Distribution2. The Gauss Sums as Universal Distributions3. The L-function at s = 04. The p-adic Partial Zeta FunctionAPPENDIX BY KARL RUBINThe Main Conjecture Introduction1. Setting and Notation2. Properties of Kolyvagin's "Euler System"3. An Application of the Chebotarev Theorem4. Example: The Ideal Class Group of5. The Main Conjecture6. Tools from lwasawa Theory7. Proof of Theorem 5.18. Other Formulations and Consequences of the Main ConjectureBibliographyIndex

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