Class field theory

Chapter 5: The First Fundamental Inequality
1. Statement of the First Inequality
2. First Inequality in Function Fields
3. First Inequality in Global Fields
4. Consequences of the First Inequality
Chapter 6: The Second Fundamental Inequality
1. Statement and Consequences of the Inequality
2. Kummer Theory
3. Proof in Kummer Fields of Prime Degree
4. Proof in p-extensions
5. Infinite Divisibility of the Universal Norms
6. Sketch of the Analytic Proof of the Second Inequality
Chapter 7: Reciprocity Law
1. Introduction
2. Reciprocity Law over the Rationals
3. Reciprocity Law
4. Higher Cohomology Groups in Global Fields
Chapter 8: The Existence Theorem
1. Existence and Ramification Theorem
2. Number Fields
3. Function Fields
4. Decomposition Laws and Arithmetic Progressions
Chapter 9: Connected Component of Idèle Classes
1. Structure of the Connected Component
2. Cohomology of the Connected Component
Chapter 10: The Grunwald–Wang Theorem
1. Interconnection between Local and Global m-th Powers
2. Abelian Fields with Given Local Behavior
3. Cyclic Extensions
Chapter 11: Higher Ramification Theory
1. Higher Ramification Groups
2. Ramification Groups of a Subfield
3. The General Residue Class Field
4. General Local Class Field Theory
5. The Conductor
Chapter 12: Explicit Reciprocity Laws
1. Formalism of the Power Residue Symbol
2. Local Analysis
3. Computation of the Norm Residue Symbol in Certain Local Kummer Fields
4. The Power Reciprocity Law
Chapter 13: Group Extensions
1. Homomorphisms of Group Extensions
2. Commutators and Transfer in Group Extensions
3. The Akizuki–Witt Map v: H²(G, A) → H²(G/H, Aᴴ)
4. Splitting Modules and the Principal Ideal Theorem
Chapter 14: Abstract Class Field Theory
1. Formations
2. Field Formations. The Brauer Groups
3. Class Formations; Method of Establishing Axioms
4. The Main Theorem
5. Exercise
6. The Reciprocity Law Isomorphism
7. The Abstract Existence Theorem
Chapter 15: Weil Groups

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 studies and illustrations of classical subjects.