Computational class field theory

Summary

Class field theory furnishes an intrinsic description of the abelian extensions of a number field which is in many cases not of an immediate algorithmic nature. We outline the algorithms available for the explicit computation of such extensions.

Class field theory is a twentieth century theory describing the set of finite abelianextensions L of certain base fields K of arithmetic type. It provides a canonical description of the Galois groups Gal.(L/K) in terms of objects defined ‘inside K’, and gives rise to an explicit determination of the maximal abelian quotient G^ab_K of the absolute Galois group G_K of K. In the classical examples, K is either a global field, that is, a number field or a function field in one variable over a finite field, or a local fieldobtained by completing a global field at one of its primes. In this paper, which takes an algorithmic approach, we restrict to the fundamental case in which the base field K is a number field. By doing so, we avoid the complications arising for p-extensions in characteristic p > 0.