  We report on a database of field extensions of the rationals, its properties
and the methods used to compute it. At the moment the database encompasses
roughly 100,000 polynomials generating distinct number fields over the
rationals, of degrees up to 15. It contains polynomials for all transitive
permutation groups up to that degree, and even for most of the possible
combinations of signature and Galois group in that range. Moreover, whenever
these are known, the fields of minimal discriminant with given group and
signature have been included.
  The database can be downloaded from
www.iwr.uni-heidelberg.de/iwr/compalg/minimum/minimum.html or from
www.mathematik.uni-kassel.de~malle/minimum/minimum.html and accessed via the
computer algebra system Kant. One of the aims of the compilation of this
database was to test the limitations of current methods for the realization of
groups as Galois groups. It turned out that these methods have limitations if
the signature of the resulting Galois extension is also prescribed.
