Computing Defining Polynomials Using Kummer Theory

Class field theory deals with Abelian extensions of base fields. It gives complete answers to the existence of Abelian extensions with given relative or absolute discriminants. However, the algorithmic construction of these extensions is not completely straightforward. There are several ways to do this, but at present the most efficient general method is the use of Kummer extensions. In the next chapter, we will describe two other methods using analytic techniques, one using Stark units and Stark’s conjecture, the other using complex multiplication. Both of these methods impose restrictions on the base field, but when they are applicable they are much more efficient.