Nilpotent extensions of number fields with bounded ramification (Q5929042)

This paper studies a variant of the inverse problem of Galois theory and Abhyankar's conjecture. If \(p\) is an odd rational prime and \(G\) is a finite \(p\)-group generated by \(s\) elements, \(s\) minimal, does there exist a normal extension \(L/{\mathbb Q}\) such that \(\text{Gal}(L/{\mathbb Q})\cong G\) with at most \(s\) rational primes that ramify in \(L\)? Given a nilpotent group \(G\) of odd order with \(s\) generators, it is obtained a Galois extension \(L/{\mathbb Q}\) with precisely \(s\) prime divisors of \({\mathbb Q}\) ramified. Furthermore if \(K\) is a number field satisfying \(K\cap{\mathbb Q}(\zeta_{p_i^{n_i}})={\mathbb Q}\) for each rational prime \(p_{i}\), such that \(p_{i}^{n_i}\mid o(G)\), \(p_{i}^{n_i+1}\nmid o(G)\), and such that there exists a rational prime \(q\) inert in \(K/{\mathbb Q}\), it is obtained a Galois extension \(E/K\) with precisely \(s\) prime divisors of \(K\) ramified. An adaptation of the Scholz-Reichardt method for the embedding problem is the main tool.