Ph.D Thesis
Ph.D Student Daniel Rabayev Upper Bound on the Minimal Number of Ramified Primes for Solvable Groups over the Rationals Department of Mathematics Full Professor Sonn Jack

Abstract

Let G be a finite group and let ramt(G) denote the minimal positive integer n such that G can be realized as the Galois group of a tamely ramified extension of Q ramified only at n finite primes. Let d(G) denote the minimal non negative integer for which there exists a subset X of G with d(G) elements such that the normal subgroup of G generated by X is all of G. It is known that d(G)<=ramt(G). However, it is unknown whether or not every finite group G can be realized as a Galois group of a tamely ramified extension of Q with exactly d(G) ramified primes.

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 ln with no more than n ramified primes. Later, Plans improved this upper bound, for odd prime l, by giving an upper bound equal to the sum of the ranks of the factors of the lower central series of G. Kisilevsky, Neftin and Sonn proved that d(G)=ramt(G) for semiabelian nilpotent groups (including groups of even order). However, not every finite l-group is a semiabelian l-group. In the case of a general odd order solvable group or a finite 2-group, there are no other known results.

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|).