On the minimal ramification problem for semiabelian groups

Abstract

It is now known that for any prime p and any finite semiabelian p-group G, there exists a (tame) realization of G as a Galois group over the rationals ‫ޑ‬ with exactly d = d(G) ramified primes, where d(G) is the minimal number of generators of G, which solves the minimal ramification problem for finite semiabelian p-groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups G. Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.

Sieves and the Minimal Ramification Problem

“…For solvable groups G, one can use Approach II, to obtain upper bounds on m(G) and for some subclasses of solvable groups, the full conjecture, see [2,15,16,20,23,24]. For example, Kisilevsky, Neftin, and Sonn [15] establish the conjecture for semi-abelian p-groups. However, to date, the conjecture is widely open for p-groups.…”

Decomposition types in minimally tamely ramified extensions of $$\mathbb {Q}$$ Q

“…This theorem shows that every finite abelian group arises as the Galois group of a tamely ramified extension of with a minimal set of ramifying primes (which as mentioned in Sect. 1 is already an easy special case of the results in [7] and [8]), with the added feature that the inertia groups for the ramifying primes can be taken to be any collection of cyclic subgroups that minimally generate G , and where the ramified primes are otherwise completely split in the extension.…”

“…It follows that if a finite group G can be realized as the Galois group of a number field K over having only tame ramification, then is the smallest possible number of primes p that are ramified in K . In [7] and [8] it is shown that all finite nilpotent semi-abelian groups can be realized by such a minimally tamely ramified extension over …”

On semiabelian p-groups

“…The family of semiabelian groups has appeared in many forms in problems that arise from field theory (e.g. geometric realizations [3], [7], [9], generic extensions [9] and the minimal ramification problem [5]). The following notion of a decomposition is used in [3] to characterize semiabelian groups: Definition 2.1.…”

“…Given a p-group G it is an open problem to find the minimal number of primes ramified in a G-extension of Q (see [8]). As a consequence of Minkowski's Theorem this number is greater or equal to d(G), the minimal number of generators of G. In [5], Kisilevsky and Sonn proved this number is exactly d(G) for a family of p-groups denoted by G p and defined as follows: Definition 1.1. Let G p be the minimal family that satisfies:…”