### School of Mathematics and Statistics

### Claus Fieker

TU Berlin

### On the Computation of Class Fields

*Friday 3rd April, 12-1pm, Carslaw 273.*
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.