Class Formation - Examples of Class Formations

Examples of Class Formations

The most important examples of class formations (arranged roughly in order of difficulty) are as follows:

• Archimedean local class field theory: The module A is the group of non-zero complex numbers, and G is either trivial or is the cyclic group of order 2 generated by complex conjugation.
• Finite fields: The module A is the integers (with trivial G-action), and G is the absolute Galois group of a finite field, which is isomorphic to the profinite completion of the integers.
• Local class field theory of characteristic p>0: The module A is the separable algebraic closure of the field of formal Laurent series over a finite field, and G is the Galois group.
• Non-archimedean local class field theory of characteristic 0: The module A is the algebraic closure of a field of p-adic numbers, and G is the Galois group.
• Global class field theory of characteristic p>0: The module A is the union of the groups of idele classes of separable finite extensions of some function field over a finite field, and G is the Galois group.
• Global class field theory of characteristic 0: The module A is the union of the groups of idele classes of algebraic number fields, and G is the Galois group of the rational numbers (or some algebraic number field) acting on A.

It is easy to verify the class formation property for the finite field case and the archimedean local field case, but the remaining cases are more difficult. Most of the hard work of class field theory consists of proving that these are indeed class formations. This is done in several steps, as described in the sections below.