A database for field extensions of the rationals
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.