Cyclotomic Fields I and IISpringer Science & Business Media, 06.12.2012 - 436 Seiten 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 11] 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. |
Inhalt
1 | |
6 | |
Relations in the Ideal Classes | 14 |
Jacobi Sums as Hecke Characters | 16 |
Gauss Sums over Extension Fields | 20 |
Application to the Fermat Curve | 22 |
CHAPTER 2 | 31 |
Bernoulli Numbers | 32 |
The Maximal pabelian pramified Extension of the Cyclotomic Zpextension | 152 |
Cyclotomic Units as a Universal Distribution | 157 |
CHAPTER 8 | 159 |
The IwasawaLeopoldt Theorem and the KummerVandiver Conjecture | 160 |
CHAPTER 7 | 166 |
The Logarithm | 171 |
Projective Limit of the Unit Groups | 175 |
A Basis for Ux over A | 179 |
Integral Stickelberger Ideals 4 | 43 |
General Comments on Indices | 48 |
The Index for k Even 6 | 49 |
The Index for k Odd 26 2225 27 32 43 48 49 | 50 |
Twistings and Stickelberger Ideals | 51 |
Stickelberger Elements as Distributions | 53 |
Universal Distributions | 57 |
The DavenportHasse Distribution | 61 |
Appendix Distributions | 65 |
CHAPTER 3 | 69 |
Primitive Lseries | 72 |
Decomposition of Lseries | 75 |
The +1eigenspaces | 81 |
Cyclotomic Units | 84 |
The Dedekind Determinant | 89 |
Bounds for Class Numbers | 91 |
CHAPTER 4 | 94 |
Measures and Power Series | 95 |
Operations on Measures and Power Series | 101 |
The Mellin Transform and padic Lfunction | 105 |
Appendix The padic Logarithm | 111 |
The padic Regulator | 112 |
The Formal Leopoldt Transform | 115 |
The padic Leopoldt Transform | 117 |
CHAPTER 5 | 123 |
The Iwasawa Algebra | 124 |
Weierstrass Preparation Theorem | 129 |
Modules over ZX | 131 |
Zpextensions and Ideal Class Groups | 137 |
The Maximal pabelian pramified Extension | 143 |
The Galois Group as Module over the Iwasawa Algebra | 145 |
CHAPTER 6 | 148 |
The CoatesWiles Homomorphism | 182 |
The Closure of the Cyclotomic Units | 186 |
CHAPTER 9 | 220 |
5 | 227 |
6 | 238 |
CHAPTER 10 | 244 |
3 | 254 |
Washingtons Theorem | 265 |
3 | 276 |
The Associated Analytic Function on the Formal | 286 |
Divisibility of Ideal Class Numbers | 295 |
The lprimary Part in an Extension of Degree Prime to | 304 |
Examples of Iwasawa | 310 |
The ArtinHasse Power Series | 319 |
The Gamma Function and Gauss Sums | 329 |
The Frobenius Endomorphism | 336 |
Eigenvalues of the Frobenius Endomorphism and the padic | 343 |
CHAPTER 16 | 360 |
The ArtinSchreier Equation | 369 |
The Frobenius Endomorphism | 378 |
The Gauss Sums as Universal Distributions | 385 |
The padic Partial Zeta Function | 391 |
APPENDIX BY KARL RUBIN | 397 |
The Ideal Class Group of Qµ | 403 |
Proof of Theorem 5 1 | 411 |
421 | |
51 | 425 |
57 | 427 |
428 | |
431 | |
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Häufige Begriffe und Wortgruppen
abelian analytic function associated power series assume automorphism B₁ Banach basis Banach space Bernoulli numbers Chapter class field theory class number coefficients completely concludes the proof conductor congruence Corollary cyclic cyclotomic fields cyclotomic units define denote det(I divisible Dwork eigenspace elements endomorphism extension factor finite number follows formal group formula Frobenius Frobenius endomorphism Galois group Gauss sums group ring Hence homomorphism ideal class group isomorphism Iwasawa algebra Kummer Leopoldt Let F Math maximal ideal module multiplicative group norm notation number field p-abelian p-adic p-adic L-function p-primary p-unit polynomial positive integer power series associated prime number primitive projective limit proves the lemma proves the theorem quasi-isomorphism rational function root of unity satisfies shows Stickelberger subgroup suffices to prove Suppose surjective Theorem 3.1 trivial unique unramified write Z,-extension