Class Field Theory: From Theory to Practice (Springer Monographs in Mathematics)

Class Field Theory: From Theory to Practice (Springer Monographs in Mathematics)


Yazar Georges Gras
Yayınevi Springer
ISBN 9783540441335
Baskı yılı 2005
Sayfa sayısı 528
Ağırlık 0,91 kg
Edisyon 1
Stok durumu Tükendi   

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.
Preface
Introduction to Global Field Theory 1
I Basic Tools and Notations 7
1 Places of K 9
2 Embeddings of a Number Field in its Completions 12
3 Number and Ideal Groups 21
4 Idele Groups - Generalized Class Groups 27
5 Reduced Ideles - Topological Aspects 45
6 Kummer Extensions 54
II Reciprocity Maps - Existence Theorems 65
1 The Local Reciprocity Map - Local Class Field Theory 65
2 Idele Groups in an Extension L/K 91
3 Global Class Field Theory: Idelic Version 104
4 Global Class Field Theory: Class Group Version 125
5 Ray Class Fields - Hilbert Class Fields 143
6 The Hasse Principle - For Norms - For Powers 176
7 Symbols Over Number Fields - Hilbert and Regular Kernels 195
III Abelian Extensions with Restricted Ramification - Abelian Closure 221
1 Generalities on H[subscript T][superscript S] / H[superscript S] and its Subextensions 221
2 Computation of A[subscript T][superscript S]:= Gal(H[subscript T][superscript S](p)/K) and T[subscript T][superscript S]:= tor[subscript Z[subscript p]] (A[subscript T][superscript S]) 240
3 Compositum of the S-split Z[subscript p]-Extensions - The p-Adic Conjecture 258
4 Structure Theorems for the Abelian Closure of K 274
5 Explicit Computations in Incomplete p-Ramification 342
6 Initial Radical of the Z[subscript p] - Extensions 348
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