Class field theory

Cover......Page 1
CLASS FIELD THEORY......Page 2
QA247.A75 2008 512.7'4„dc22......Page 3
Contents......Page 4
Preface to the New Edition......Page 6
Preface......Page 8
1. Idéles and Idèle Classes......Page 9
2. Cohomology......Page 11
3. The Heitrand Quotient......Page 13
4. Local Class Field Theory......Page 16
2. FIrst Inequality In Function Fields......Page 19
3. First Inequality in Global Fields......Page 21
4. Consequences of the First Inequality......Page 24
1. Statement and Consequences of the Inequality......Page 27
2. Kuinmer Theory......Page 29
3. Proof in Kummer Fields of Prime Degree......Page 32
4. Proof in p-extensions......Page 35
5. Infinite Divisibility of the Universal Norms......Page 40
6. Sketch of the Analytic Proof of the Second Inequality......Page 41
1. Introduction......Page 43
2. Reciprocity Law over the Rationals......Page 44
3. Reciprocity Law......Page 49
4. Higher Cobomology Groups in Global Fields......Page 60
1. Existence and Ramification Theorem......Page 63
2. Number Fields......Page 64
3. Function Fields......Page 67
4. Decomposition Laws and Arithmetic Progressions......Page 70
1. Structure of the Connected Component......Page 73
2. Cohomology of the Connected Component......Page 78
1. Interconnection between Local and Global rn-tb Powers......Page 81
2. Abelian Fields with Given Local Behavior......Page 84
3. Cyclic Extensions......Page 89
1. Higher Ramification Groups......Page 91
2. Ramification Groups of a Subfield......Page 94
3. The General Residue Class Field......Page 98
4. General Local Class Field Theory......Page 100
5. The Conductor......Page 107
Appendix: Induced Characters......Page 112
1. Formalism of the Power Residue Symbol......Page 117
2. Local Analysis......Page 119
3. Computation of the Norm Residue Symbol in Certain Local Kummer Fields......Page 122
4. The Power Reciprocity Law......Page 130
1. HomomorphIsms of Group Extensions......Page 135
2. Commutators and bansfer in Group Extensions......Page 139
3. The Akisuki-Witt Map......Page 142
4. Splitting Modules and the Principal Ideal Theorem......Page 145
1. Fbrmations......Page 151
2. Field Fbrmations. The Brauer Groups......Page 154
3. Class Formations; Method of Establishing Axioms......Page 158
Exercise......Page 165
5. The Reciprocity Law isomorphism......Page 166
6. The Abstract Existence Theorem......Page 171
CHAPTER XV  Weil Groups......Page 175
Bibliography......Page 199