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They have been substantially revised and expanded from an earlier version, based on my notes from v2. My approach to class field theory in these notes is eclectic. Although it is possible to prove the main theorems in class field theory using neither analysis nor cohomology, there are major theorems that can not even be stated without using one or the other, for example, theorems on densities of primes, or theorems about the cohomology groups associated with number fields.

When it sheds additional light, I have not hesitated to include more than one proof of a result. The heart of the course is the odd numbered chapters. Except for its first section, explicit local class field theory. Reprintedby Gordon and Breach, Artin, E. Cassels, J. Fesenko, I. Goldstein, L. Iyanaga, S. Janusz, G. Koch, H. Ency- clopaedia of Mathematical Sciences, Vol. Lang, S. Neukirch, J. Serre, J-P. You may make one copy of these notes for your own personal use.

Books including an introduction to class field theory Cohn, H. Sources for the history of algebraic number theory and class field theory Edwards, H. Hermann, Paris, , pp — Reprinted by Chelsea, New York, Appendi x2 in Iyanaga We c. Stevenhagan and H. Intelligencer, Chapter I. Local Class Field Theory 9 1. Statements of the Main Theorems 9 Consequences of Theorems 1.

Construction of the extension of. The Cohomology of Groups 1. Local Class Field Theory Continued 77 1. Introduction 77 2. The Cohomology of Unramified Extensions 80 The cohomology of the units, 80; The invariant map, 81; Computation of the local Artin map, 83; 3. The Cohomology of Ramified Extensions 85 4. Brauer Groups 91 1. Simple Algebras; Semisimple Modules 91 Semisimple modules, 91; Simple k -algebras, 93; Modules over simple k -algebras, 95; 2. Definition of the Brauer Group 96 Tensor products of algebras, 96; Centralizers in tensor products, 97; Primordial elements, 97; Simplicity of tensor products, 98; The Noether-Skolem Theorem, 99; Definition of the Brauer group, ; Extension of the base field, ; 3.

The Brauer Group and Cohomology Maximal subfields, ; Central simple algebras and 2-cocycles, ; 4. Complements Semisimple algebras, ; Algebras, cohomology, and group extensions, ; Brauer K -theory, ; Notes, ; groups and Chapter V.

Global Class Field Theory: Statements 1. Dirichlet gressions 3. L -Series and the Density of Primes 1. Dirichlet series and Euler products 2. Convergence Results Dirichlet series, ; Euler products, ; Partial zeta functions; the residue for- mula, ; 3. Density of the Prime Ideals Splitting in an Extension 4. Global Class Field Theory: Proofs 1. Outline 2. The Cohomology of the Units 4. The Algebraic Proof of the Second Inequality 7. Application to the Brauer Group 8.

Completion of the Proof of the Reciprocity Law 9. The Existence Theorem Complements 1. The Classification of Quadratic Forms over a Number Field Generalities on quadratic forms, ; The local-global principle, ; The classifi- cation of quadratic forms over a local field, ; Classification of quadratic forms over global fields, ; Applications, ; 5. Density Theorems 6. Function Fields 7. Cohomology of Number Fields 8.

For abelian extensions, the theory was developed between roughly and by Kronecker, Weber, Hilbert, Takagi, Artin, and others. For nonabelian extensions, serious progress began only about 25 years ago with the work of Langlands. Today, the nonabelian theory is in roughly the state that abelian class field theory was in years ago: there are comprehensive conjectures but few proofs. In this course, we shall be concerned only with abelian class field theory.

The proof of the general case is left as an exercise. I Classification of unramified abelian extensions. The corresponding C primes are called finite, real, and comple xrespectively. Each 5 b c positive integers. A class field exists for each subgroup H of C 0. Moreover, every finite abelian extension C m some m and H. Theorem 0. C L The class field for the trivial subgroup of for. The field p , No other prime ramifies, and so K [ does not ramify in p K p i i i unramified over. The above construction shows that group of K C C 2.

Koch , 2. In particular, we see that, by using class field theory, it is easy to construct 2 Q such that C : C quadratic extensions of is very large.

All methods of constructing ele- no quadratic field was known with : C C ments of order 3 in the class groups of quadratic number fields seem to involve elliptic curves. Artin did show that there is a natural isomorphism, but even he did this to obtain another result that interested him. In this way, we obtain an isomorphism P. Let L be an abelian extension of K ,andlet Theorem be the set of finite S primes ramifying in.

If the map were not surjective, then there would be a proper extension of K in which K every prime of splits. It is easy to see analytically much harder algebraically that no such extension exists. Let m be a positive integer that is either odd or divisible by 4, and Example 0. Inthe12 of his famous problems, Hilbert asked whether the ray class fields for other number fields can be generated by the special values of explicit holomorphic functions.

Nonabelian class field theory. For nonabelian ex- S tensions, this is no longer true, and any description of the sets must be analytic. Thus, the conjecture answers the original question for all finite Galois extensions of Q. Let 0. From a different perspective, it describes the local components of the global Artin map. By a local field, I mean a field K that is locally compact with respect to a nontrivial valuation. Thus it is Q a a finite extension of p ; for some p F b a finite extension of the field of Laurent series over the field with T p p elements; or R c C archimedean case.

A generator of m is called a prime element of K rather than the more customary U K local uniformizing parameter. K extensions to 1. Statements of the Main Theorems K The composite of two finite abelian extensions of is again a finite abelian exten- al ab isan in of all finite abelian extensions of K K sion of.

See the appendi xto this chapter for a review of the Galois theory of infinite extensions. Let be a finite unramified extension of K. For any nonarchimedian local field, Theorem 1. Asubgroup N of K Theorem 1. The proofs of these theorems will occupy most of the rest of this chapter and of chapter 3.

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