Jürgen Neukirch

Class field theory

I. Group and Field Theoretic Foundations.- 1. Infinite Galois Theory.- 2. Profinite Groups.- 3. G-Modules.- 4. The Herbrand Quotient.- 5. Kummer Theory.- 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.- 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 Qp.- 5. The Hilbert Symbol.- 6. Formal Groups.- 7. Fields of ?n-th Division Points.- 8. Higher Ramification Groups.- 9. The Weil Group.- IV. Global Class Field Theory.- 1. Algebraic Number 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.- 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.

The p -Adic Numbers

The p-adic numbers were invented at the beginning of the twentieth century by the German mathematician Kurt Hensel (1861–1941). The aim was to make the methods of power series expansions, which play such a dominant role in the theory of functions, available to the theory of numbers as well. The idea sprang from the observation that numbers behave in many ways just like functions, and in a certain sense numbers may also be regarded as functions on a topological space.

Part III Global Class Field Theory

We assume the reader is familiar with the basic concepts and theorems of algebraic number theory for which we refer to the standard text books, for example, [6], [21], [30]. Nevertheless, in this section we briefly summarize the for us most important facts.

Part I Cohomology of Finite Groups

The cohomology of finite groups deals with a general situation that occurs frequently in different concrete forms. For example, if L|K is a finite Galois extension with Galois group G, then G acts on the multiplicative group L × of the extension field L. In the special case of an extension of finite algebraic number fields, G acts on the ideal group J of the extension field L. The theory of group extensions provides us with the following example: If G is an abstract finite group and A is a normal subgroup, then G acts on A via conjugation. In representation theory we study matrix groups G that act on a vector space. The basic notion underlying all these examples is that of a G-module. We will now present some general considerations about G-modules, some of which the reader may already know from the theory of modules over general rings.

Part II Local Class Field Theory

Local and global class field theory, as well as a series of further theories for which the name class field theory is similarly justified, have the following principle in common. All of these theories involve a canonical bijective correspondence between the abelian extensions of a field K and certain subgroups of a corresponding module AK associated with the field K.

Cohomology of Local Fields

We now begin the development of cohomology in number theory. As a ground field we take a nonarchimedean local field k, i.e. a field which is complete with respect to a discrete valuation and has a finite residue field. This covers two cases, namely p-adic local fields, i.e. finite extensions of \(\mathbb{Q}_{p}\) for some prime number p, and fields of formal Laurent series \(\mathbb{F}((t))\) in one variable over a finite field. For the basic properties of local fields we refer to [160], chapters II and V. As always, \(\bar{k}|k\) denotes a separable closure of k and K|k the subextensions of \(\bar{k}|k\). v k denotes the valuation of k, normalized by \(v_{k}(k^{\times}) = \mathbb{Z}\), and κ the residue field. For every Galois extension K|k we set

Duality Properties of Profinite Groups

H^i(K|k) := H^i(G(K|k),K^{\times}),\quad i \geq 0.

Some Homological Algebra

If K|k is finite, we also set \(\hat{H}^{i}(K|k) = \hat{H}^{i}(G(K|k),K^{\times})\) for \(i \in \mathbb{Z}\). The basis for the results in this chapter is the following theorem.