Class field theory

This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians. Title page......Page 1 Date-line......Page 2 Annotation......Page 3 Acknowledgment......Page 5 Table of contents......Page 7 Ideles and Idele Classes......Page 9 Cohomology......Page 12 The Herbrand Quotient......Page 16 Local Class Field Theory......Page 20 CLASS FIELD THEORY......Page 25 1. Statement of the first inequality......Page 26 2. First inequality in function fields......Page 27 3. First inequality in global fields......Page 29 4. Consequences of the first inequality......Page 34 1. Statement and consequences of the inequality......Page 38 2. Kummer theory......Page 41 3. Proof in Kummer fields of prime degree......Page 46 4. Proof in $p$-extensions......Page 51 5. Infinite divisibility of universal norms......Page 59 1. Introduction......Page 61 2. Reciprocity law over the rationals......Page 62 3. The reciprocity law......Page 71 4. Higher dimensional cohomology groups......Page 89 1. Exlitence and ramification theorem!......Page 92 2. Number field......Page 93 3. Function fields......Page 98 4. Decomposition laws and arithmetic progressions......Page 102 1. Structure of the connected component......Page 104 2. Cohomology of the connected component......Page 112 1. Interconnection between global and local theory......Page 115 2. Abelian fields with given local behavior......Page 120 3. Cyclic extensions......Page 127 1. Higher ramification groups......Page 130 2. Ramification groups of a subfield......Page 135 3. The general residue class field......Page 142 4. General local class field theory......Page 145 5. The conductor......Page 155 Appendix: Induced characters......Page 165 1. Formalism of the power residue symbol......Page 171 2. Local analysis......Page 173 3. Computation of the norm residue symbol In certain local Kummer fields......Page 178 4. Power reciprocity law......Page 190 1. Homomorphlsma of group extensions......Page 196 2. Commutators and transfers......Page 202 3. The map v: $H^2(G,A)\\to H^2(G/H,A^H)$......Page 207 4. Splitting modules and Principal Ideal Theorem......Page 211 1. Formations......Page 219 2. Field formations, Brauer group......Page 224 3. Class formations, method for establishing the axioms......Page 231 4. Main theorem......Page 237 5. Reciprocity law isomorphisms......Page 244 6. Abstract existence theorem......Page 253 Chapter Fifteen: Weil groups......Page 258