Class Field Theory

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