Class Field Theory (Grundlehren der mathematischen Wissenschaften, 280)

Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper­ spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol­ ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group­ theoretical principle, admitting an entirely elementary description. This de­ scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep­ tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here. Cover Title page Preface Chapter 1. Group and Field Theoretic Foundations 1. Infinite Galois Theory 2. Profinite Groups 3. G-Modules 4. The Herbrand Quotient 5. Kummer Theory Chapter II. General Class Field Theory 1. Frobenius Elements and Prime Elements 2. The Reciprocity Map 3. The General Reciprocity Law 4. Class Fields 5. Infinite Extensions Chapter III. Local Class Field Theory 1. The Class Field Axiom 2. The Local Reciprocity Law 3. Local Class Fields 4. The Norm Residue Symbol over Q_p 5. The Hilbert Symbol 6. Formal Groups 7. Fields of π^n-th Division Points 8. Higher Ramification Groups 9. The Weil Group Chapter IV. Global Class Field Theory 1. Algebraic N umber Fields 2. Ideles and Idele Classes 3. Galois Extensions 4. Kummer Extensions 5. The Class Field Axiom 6. The Global Reciprocity Law 7. Global Class Fields 8. The Ideal- Theoretic Formulation of Class Field Theory 9. The Reciprocity Law of Power Residues Chapter V. Zeta Functions and L-Series 1. The Riemann Zeta Function 2. The Dedekind Zeta Function 3. The Dirichlet L-Series 4. The Artin L-Series 5. The Equality of Dirichlet L-Series and Artin L-Series 6. Density Theorems Literature Index The present manuscript is an improved edition of a text that first appeared under the same title in Bonner Mathematische Schriften, no.26, and originated from a series of lectures given by the author in 1965/66 in Wolfgang Krull's seminar in Bonn. Its main goal is to provide the reader, acquainted with the basics of algebraic number theory, a quick and immediate access to class field theory. This script consists of three parts, the first of which discusses the cohomology of finite groups. The second part discusses local class field theory, and the third part concerns the class field theory of finite algebraic number fields.