Minimal Ramification in Nilpotent Extensions

TL;DR

This paper establishes bounds on the minimal number of ramified primes in nilpotent Galois extensions of number fields, extending previous results and confirming a conjecture for certain central extensions.

Contribution

It introduces a method to bound ramification in nilpotent extensions using central group extensions, improving upon prior bounds and confirming Boston's conjecture in specific cases.

Findings

Provides an upper bound for ramified primes in nilpotent extensions

Sharpens previous results by Geyer, Jarden, and Plans

Confirms Boston's conjecture for certain central extensions

Abstract

Let $G$ be a finite nilpotent group and $K$ a number field with torsion relatively prime to the order of $G$. By a sequence of central group extensions with cyclic kernel we obtain an upper bound for the minimum number of prime ideals of $K$ ramified in a Galois extension of $K$ with Galois group isomorphic to $G$. This sharpens and extends results of Geyer and Jarden and of Plans. Also we confirm Boston's conjecture on the minimum number of ramified primes for a family of central extensions by the Schur multiplicator.