ANTIC: Explicit field construction


This projects aims to turn Shafarevich’s theorem on the existence of solvable number fields into an algorithm: Given a solvable group, we will construct field extensions of the rational numbers having this group as Galois group. The fields will be constructed as towers of cyclic fields via class field theory. Galois cohomology will be used to analyse the embedding problems and to control the Galois group of the tower of number fields. Finally, large tables of fields with a fixed Galois group will be computed to study the asymptotic distribution of arithmetic invariants, including class numbers (heuristics of Cohen-Lenstra, Malle) and discriminants (Malle’s conjecture).