Author | Lang, Serge. author |
---|---|
Title | Cyclotomic Fields [electronic resource] / by Serge Lang |
Imprint | New York, NY : Springer New York, 1978 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-9945-5 |
Descript | 253p. online resource |
1 Character Sums -- 1. Character Sums Over Finite Fields -- 2. Stickelbergerโs 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 (ยฑ1)-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