Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book. Global class field theory is a major achievement of algebraic number theory, based on the functorial properties of the reciprocity map and the existence theorem. The author works out the consequences and the practical use of these results by giving detailed studies and illustrations of classical subjects (classes, idèles, ray class fields, symbols, reciprocity laws, Hasse's principles, the Grunwald-Wang theorem, Hilbert's towers,...). He also proves some new or less-known results (reflection theorem, structure of the abelian closure of a number field) and lays emphasis on the invariant (/cal T) p, of abelian p-ramification, which is related to important Galois cohomology properties and p-adic conjectures. This book, intermediary between the classical literature published in the sixties and the recent computational literature, gives much material in an elementary way, and is suitable for students, researchers, and all who are fascinated by this theory.
In the corrected 2nd printing 2005, the author improves some mathematical and bibliographical details and adds a few pages about rank computations for the general reflection theorem; then he gives an arithmetical interpretation for usual class groups, and applies this to the Spiegelungssatz for quadratic fields and for the p-th cyclotomic field regarding the Kummer--Vandiver conjecture in a probabilistic point of view.
(Table of content)
Preface Introduction to Global Class Field Theory Chapter I: Basic Tools and Notations 1) Places of a number field 2) Embeddings of a Number Field in its Completions 3) Number and Ideal Groups 4) Idele Groups - Generalized Class Groups 5) Reduced Ideles - Topological Aspects 6) Kummer Extensions Chapter II: Reciprocity Maps - Existence Theorems 1) The Local Reciprocity Map - Local Class Field Theory 2) Idele Groups in an Extension L/K 3) Global Class Field Theory: Idelic Version 4) Global Class Field Theory: Class Group Version 5) Ray Class Fields 6) The Hasse Principle - For Norms - For Powers 7) Symbols Over Number Fields - Hilbert and Regular Kernels Chapter III: Abelian Extensions with Restricted Ramification - Abelian Closure 1) Generalities on H(T)/H and its Subextensions 2) Computation of A(T) := Gal(H(T)/K) and T(T) := tor(A(T)) 3) Study of the compositum of the Zp-extensions - The p-adic Conjecture 4) Structure Theorems for the Abelian Closure of K 5) Explicit Computations in Incomplete p-Ramification 6) The Radical of the Maximal Elementary Subextension of the compositum of the Zp-extensions Chapter IV: Invariant Classes Formulas in p-ramification - Genus Theory 1) Reduction to the Case of p-Ramification 2) Injectivity of the Transfer Map: A(K,p) to A(L,p) 3) Determination of invariant classes of A(L,p) and T(L,p) - p-Rationality 4) Genus Theory with Ramification and Decomposition Chapter V: Cyclic Extensions with Prescribed Ramification 1) Study of an Example 2) Construction of a Governing Field 3) Conclusion and Perspectives Appendix: Arithmetical Interpretation of the second cohomology group of G(T,S) over Zp 1) A General Approach by Class Field Theory 2) Complete p-Ramification Without Finite Decomposition 3) The General Case - Infinitesimal Knot Groups Bibliography Index of Notations