Ph.D Thesis | |

Ph.D Student | Rabayev Daniel |
---|---|

Subject | Upper Bound on the Minimal Number of Ramified Primes for Solvable Groups over the Rationals |

Department | Department of Mathematics |

Supervisor | PROFESSOR EMERITUS Jack Sonn |

Let* G* be a finite group and
let *ram ^{t}(G) *denote the minimal positive integer

Let *G* be a finite group. Serre
found that for odd prime* l*, the Scholz-Reichardt method yields
realizations of an *l*-group of order *l ^{n}* with no more
than

We will give an upper bound on the
minimal number of ramified primes for all finite 2-groups and for all odd order
solvable groups. Like previous results, the upper bound will depend solely on
the structure of *G* as an abstract group. The upper bound for the
2-groups is extracted from Shafarevich's proof of realizing *l*-groups
over number fields. However, the bound is super exponential with respect to the
order of *G*. The upper bound for the odd order solvable groups is
extracted from Neukirch's proof of the realization of odd order solvable groups
over number fields. In this case, the upper bound equals *3log(|G|)*.