Algebra Seminar

Class field theory is one of the oldest and most beautiful areas of number theory. Starting with a number field k it completely classifies all Abelian extensions K/k, by proving the existence of a bijection between the Abelian extension of k and certain groups of ideals of k.

In this talk I will discuss a method for obtaining defining equations for K/k, based on an explicit realisation of the Artin-Frobenius map. Although the theory is rather old, very little progress was made on the explicit construction of those class fields. However, recently, constructive number theory and computer algebra systems have become powerful enough to allow the actual computation of defining equations for class fields of moderate size.

If k is normal this can be used to realize certain group extensions of Gal(k/Q) as the Galois group of K/Q.

Another important application is to the construction of fields with small discriminant, since the discriminant can be computed quickly independently of the computation of the defining equations.