Cyclotomic Fields

1. Front Matter

   Pages i-xi

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2. Character Sums

   • Serge Lang

   Pages 1-25

3. Stickelberger Ideals and Bernoulli Distributions

   • Serge Lang

   Pages 26-68

4. Complex Analytic Class Number Formulas

   • Serge Lang

   Pages 69-93

5. The p-adic L-function

   • Serge Lang

   Pages 94-122

6. Iwasawa Theory and Ideal Class Groups

   • Serge Lang

   Pages 123-147

7. Kummer Theory over Cyclotomic Z _p -extensions

   • Serge Lang

   Pages 148-165

8. Iwasawa Theory of Local Units

   • Serge Lang

   Pages 166-189

9. Lubin-Tate Theory

   • Serge Lang

   Pages 190-219

10. Explicit Reciprocity Laws

    • Serge Lang

    Pages 220-243

11. Back Matter

    Pages 244-256

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Kummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. For a long period in the 20th century this aspect of Kummer's work seems to have been largely forgotten, except for a few papers, among which are those by Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. In the mid 1950's, the theory of cyclotomic fields was taken up again by Iwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry, and wrote a great sequence of papers investigating towers of cyclotomic fields, and more generally, Galois extensions of number fields whose Galois group is isomorphic to the additive group of p-adic integers. Leopoldt concentrated on a fixed cyclotomic field, and established various p-adic analogues of the classical complex analytic class number formulas. In particular, this led him to introduce, with Kubota, p-adic analogues of the complex L-functions attached to cyclotomic extensions of the rationals. Finally, in the late 1960's, Iwasawa [Iw 1 I] . made the fundamental discovery that there was a close connection between his work on towers of cyclotomic fields and these p-adic L-functions of Leopoldt-Kubota.

• Fields
• Kreiskörper
• Prime
• algebra
• finite field
• homomorphism
• number theory

• Department of Mathematics, Yale University, New Haven, USA

  Serge Lang

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