Cyclotomic Fields

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. Language: English