Nilpotent extensions of number fields with bounded ramification

A. Cueto-Hernández and G.D. Villa-Salvador

Vol. 196 (2000), No. 2, 297–316

DOI: 10.2140/pjm.2000.196.297

Abstract

We study 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∕ℚ such that Gal (L∕ℚ)≅G with at most s rational primes that ramify in L? Given a nilpotent group of odd order G with s generators, we obtain a Galois extension L∕ℚ with precisely s prime divisors of ℚ ramified. Furthermore if K is a number field satisfying K ∩ ℚ(ζ_piⁿⁱ) = ℚ for each rational prime p_i, such that p_i^n_i|∘ (G), p_i^n_i+1|∕∘ (G), and such that there exists a rational prime q inert in K∕ℚ, we obtain 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 our main tool.

Milestones

Received: 9 April 1999

Revised: 11 January 2000

Published: 1 December 2000