Cyclotomic fields II. Front Cover. Serge Lang. Springer-Verlag, Cyclotomic Fields II · S. Lang Limited preview – QR code for Cyclotomic fields II. 57 CROWELL/Fox. Introduction to Knot. Theory. 58 KOBLITZ. p-adic Numbers, p- adic. Analysis, and Zeta-Functions. 2nd ed. 59 LANG. Cyclotomic Fields. In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive . New York: Springer-Verlag, doi/ , ISBN , MR · Serge Lang, Cyclotomic Fields I and II.
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Account Options Sign in. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie My library Help Advanced Book Search. Cyclotomic Fields I and II. Kummer’s work on cyclotomic fields paved the way for the cyc,otomic of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others.
However, the success of this general theory cycltomic 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], Cyclotomoc [A-H] and Vandiver [Va].
In the mid ‘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 ‘s, Iwasawa [Iw 11] made the fundamental discovery fielfs there was a close connection between his work on towers of cyclotomic fields and these p-adic L-functions of Leopoldt – Kubota.
Selected pages Title Page. Relations in the Ideal Classes. Jacobi Sums as Hecke Characters. Gauss Sums over Extension Fields. Application to the Fermat Curve. The Index of the First Stickelberger Ideal. Application of the Logarithm to the Local Symbol. Statement of the Reciprocity Laws. General Comments on Fislds. The Index for k Even.
The Index for k Odd. Twistings and Stickelberger Ideals. Stickelberger Elements as Distributions. Measures and Power Series. Operations on Measures and Power Series. The Mellin Transform and padic Lfunction.
Appendix The padic Logarithm. The Formal Leopoldt Transform.
The padic Leopoldt Transform. Zpextensions and Ideal Class Groups. The Maximal pabelian pramified Extension. Cyclotomic Units as a Universal Distribution. Iwasawa Theory of Local Units. Projective Limit of the Unit Groups. A Basis for UX over. The Closure of the Cyclotomic Units. A Local Pairing with the Logarithmic Derivative.
The Main Lemma for Highly Divisible x and 0. The Main Theorem for the Symbol x xnn. The Main Theorem for Divisible x and 0 unit. End of the Proof of the Main Theorems. Iwasawa Invariants for Measures. Application to the Bernoulli Distributions. Class Numbers as Products of Bernoulli Numbers. Basic Lemma and Applications.
reference request – Good undergraduate level book on Cyclotomic fields – Mathematics Stack Exchange
Equidistribution and Normal Families. Proof of the Basic Lemma. Measures and Power Fiels in the Composite Case. Computation of Lp1 y in the Composite Case Contents. Analytic Representation of Roots of Unity.
The Ideal Class Group of Qup. Proof of Theorem 5 1. Common terms and phrases A-module A pm assume automorphism Banach basis Banach space Bernoulli numbers Bernoulli polynomials Chapter class field theory class number CM field coefficients commutative concludes the proof conductor congruence Corollary cyclic cyclotomic fields cyclotomic units define denote det I Dirichlet character distribution relation divisible Dwork eigenspace eigenvalue elements endomorphism extension factor follows formal group formula Frobenius Frobenius endomorphism Galois group Gauss sums gives group ring Hence homomorphism ideal class group isomorphism kernel KUBERT Kummer Leopoldt Let F linear mod 7t module multiplicative group norm notation number field odd characters p-unit polynomial positive integer power series associated prime number primitive projective limit Proposition proves the lemma proves the theorem Q up quasi-isomorphism rank right-hand side root of unity satisfies shows subgroup suffices to prove Suppose surjective Theorem 3.