### A database for field extensions of the rationals, by Juergen Klueners and Gunter Malle

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.

Juergen Klueners and Gunter Malle <klueners@iwr.uni-heidelberg.de>