Cyclotomic Fields

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.

1 Character Sums.- 1. Character Sums Over Finite Fields.- 2. Stickelberger’s Theorem.- 3. Relations in the Ideal Classes.- 4. Jacobi Sums as Hecke Characters.- 5. Gauss Sums Over Extension Fields.- 6. Application to the Fermat Curve.- 2 Stickelberger Ideals and Bernoulli Distributions.- 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.- 3 Complex Analytic Class Number Formulas.- 1. Gauss Sums on Z/mZ.- 2. Primitive L-series.- 3. Decomposition of L-series.- 4. The (±1)-eigenspaces.- 5. Cyclotomic Units.- 6. The Dedekind Determinant.- 7. Bounds for Class Numbers.- 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.- 4. The p-adic Regulator.- 5. The Formal Leopoldt Transform.- 6. The p-adic Leopoldt Transform.- 5 Iwasawa Theory and Ideal Class Groups.- 1. The Iwasawa Algebra.- 2. Weierstrass Preparation Theorem.- 3. Modules over Zp[[X]].- 4. Zp-extensions and Ideal Class Groups.- 5. The Maximal p-abelian p-ramified Extension.- 6. The Galois Group as Module over the Iwasawa Algebra.- 6 Kummer Theory over Cyclotomic Zp-extensions.- 1. The Cyclotomic Zp-extension.- 2. The Maximal p-abelian p-ramified Extension of the Cyclotomic Zp-extension.- 3. Cyclotomic Units as a Universal Distribution.- 4. The Leopoldt-Iwasawa Theorem and the Vandiver Conjecture.- 7 Iwasawa Theory of Local Units.- 1. The Kummer-Takagi Exponents.- 2. Projective Limit of the Unit Groups.- 3. A Basis for U(?) over A.- 4.The Coates-Wiles Homomorphism.- 5. The Closure of the Cyclotomic Units.- 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.- 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 ? = xn.- 5. The Main Theorem for the Symbol ?x, xn?n.- 6. The Main Theorem for Divisible x and ? = unit.- 7. End of the Proof of the Main Theorems.