Finding bounds for root discriminants of number field extensions has been an area of interest for many mathematicians. The most notable bound is the Serre-Odlyzko bound, which was found in 1975. Assuming the Generalized Riemann Hypothesis, this bound gives a lower bound for the infimum limit of root discriminants of number field extensions as the degree gets large. In other words if the Generalized Riemann Hypothesis is true, for every degree there are only finitely many number field extensions with root discriminants less than this bound. It has been conjectured that there are only finitely many Galois field extensions with root discriminants less than or equal to the Serre-Odlyzko bound. The problem of identifying Galois number fields with root discriminants less than or equal to the Serre-Odlyzko bound has also been posed. Seven thousand sixty-three such Abelian fields have been found. One natural step is to look at fields with solvable Galois groups.

Class field theory and Kummer theory were used to develop a method to find such fields. There were three main challenges in developing the algorithm. The first one was to compute all cyclic extensions of a certain number field with Galois root discriminants less than the Serre-Odlyzko bound. Although it produced all the desired cyclic extensions, it also produced extensions whose Galois root discriminants were too large. The second challenge was to filter out the undesirable extensions. The third challenge was to find defining polynomials with minimal degree. Using this method, complete lists of polynomials with root discriminants less than or equal to the Serre-Odlyzko bound were found for several Galois groups. Lists had been found for all sextic groups and the completeness of the lists had been confirmed for all except two. The methods developed in this paper confirmed the completeness of their lists and generated lists of polynomials for two septic groups, several octic groups and one degree twelve group.