分圆域
2009-6
世界图书出版公司
朗
432
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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 class number formulas. In particular, this led himto introduce, with Kubota, p-adic analogues of the complex L-functionsattached to cyclotomic extensions of the rationals. Finally, in the late 1960's,Iwasawa [Iw 11] made the fundamental discovery that there was a closeconnection between his work on towers of cyclotomic fields and these p-adicL-functions of Leopoldt-Kubota. The classical results of Kummer, Stickelberger, and the Iwasawa-Leopoldttheories have been complemented by, and received new significance from thefollowing directions:
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.
作者：(美国)朗(Lang.S.)
NotationIntroductionCHAPTER 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 CurveCHAPTER 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. DistributionsCHAPTER 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 NumbersCHAPTER 4 The p-adic L-functionCHAPTER 5 Iwasawa Theory and Ideal Class GroupsCHAPTER 6 Kummer Theory over Cyclotomic ZextensionsCHAPTER 7 Iwasawa Theory of Local Units……
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