On the computation of Hilbert class fields
The authors present an algorithm for construction of \(\Gamma(k)\) the Hilbert class field of \(k\) (unramified algebraic extension with relative Galois group isomorphic to the class group \(Cl_k)\). The method is to compose Kummer extensions using intermediate fields. The idea is to adjoin \({\root p\of {\mu}}\) where \((\mu)\) is a principal ideal which is the \(p\)-power of an ideal of order \(p\). Since each such \(\mu\) can be endowed with (unknown) units, the number of possibilities to test might be astronomically large, but there are tricks for reducing the search. This extraordinary algorithm is available through KASH (KANT).
scientific article; zbMATH DE number 1202431
scientific article