On the Minimal Number of Ramified Primes in Some Solvable Extensions of Q

Given a finite group G, let ram(G) (resp. ramt(G)) denote the minimal positive integer such that G can be realized as the Galois group of an extension of Q (resp. a tamely ramified extension of Q) ramified only at ram(G) (resp. ramt(G)) finite primes. The present paper is devoted to study ram(G) and ramt(G), for some solvable groups G. More precisely, we consider the case where G is either a finite nilpotent group of odd order or a generalized dihedral group. Let l be an odd prime number. The Scholz-Reichardt’s Theorem establishes that every l-group G can be realized as the Galois group of some extension of Q [Re]. By Burnside’s Basis Theorem and Kronecker–Weber’s Theorem, ram(G) must be greater than or equal to the minimal number of generators of G. At present, it is not known whether this lower bound coincides with the exact value of ram(G) and ramt(G), although this is claimed in [Cu-He] (see Remark 2.10). A Galois extension of Q with Galois group G arises by proper resolution of a chain of central embedding problems, starting with the trivial epimorphism GQ → {1}. Moreover, if one restricts himself to embedding problems with kernel of order l, then this process can be made adding only one new ramified prime at each step. Hence, ramt(G) ≤ n, where ln is the order of G [Se, Chap. 2] (see also the generalization in [Ge-Ja], where the ground field Q is replaced by a general global field). We prove a better upper bound for ramt(G), less than or equal to the sum of the minimal number of generators of the factors in the lower central series of G. In order to obtain this improvement, we allow arbitrary cyclic kernels and we show that it still suffices to admit just one new ramified prime (for Frattini or split embedding problems). In addition,