This book is a combined edition of the books previously published as Cyclotomic Fields, Vol. I and II. It continues to provide a basic introduction to the theory of these number fields, which are of great interest in classical number theory, as well as in other areas, such as K-theory. Cyclotomic Fields begins with basic material on character sums, and proceeds to treat class number formulas, p-adic L-functions, Iwasawa theory, Lubin-Tate theory, and explicit reciprocity laws, and the Ferrero-Washington theorems, which prove Iwasawa's conjecture on the growth of the p-primary part of the ideal class group.…mehr
This book is a combined edition of the books previously published as Cyclotomic Fields, Vol. I and II. It continues to provide a basic introduction to the theory of these number fields, which are of great interest in classical number theory, as well as in other areas, such as K-theory. Cyclotomic Fields begins with basic material on character sums, and proceeds to treat class number formulas, p-adic L-functions, Iwasawa theory, Lubin-Tate theory, and explicit reciprocity laws, and the Ferrero-Washington theorems, which prove Iwasawa's conjecture on the growth of the p-primary part of the ideal class group.
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Summary of the Contents: Character Sums.- Stickelberger Ideals and Bernoulli Distributions.- Complex Analytic Class Number Formulas.- The p-adic L-function.- Iwasawa Theory and Ideal Class Groups.- Kummer Theory over Cyclotomic Zp-extensions.- Iwasawa Theory of Local Units.- Lubin-Tate Theory.- Explicit Reciprocity Laws.- Measures and Iwasawa Power Series.- The Ferrero-Washington Theorems.- Measures in the Composite Case.- Divisibility of Ideal Class Numbers.- p-adic Preliminaries.- The Gamma Function and Gauss Sums.- Gauss Sums and the Artin-Schreier Curve.- Gauss Sums as Distributions.- Appendix: The Main Conjecture.- Bibliography.- Index.
Summary of the Contents: Character Sums.- Stickelberger Ideals and Bernoulli Distributions.- Complex Analytic Class Number Formulas.- The p-adic L-function.- Iwasawa Theory and Ideal Class Groups.- Kummer Theory over Cyclotomic Zp-extensions.- Iwasawa Theory of Local Units.- Lubin-Tate Theory.- Explicit Reciprocity Laws.- Measures and Iwasawa Power Series.- The Ferrero-Washington Theorems.- Measures in the Composite Case.- Divisibility of Ideal Class Numbers.- p-adic Preliminaries.- The Gamma Function and Gauss Sums.- Gauss Sums and the Artin-Schreier Curve.- Gauss Sums as Distributions.- Appendix: The Main Conjecture.- Bibliography.- Index.
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