## Construction of unramified extensions with a prescribed solvable Galois group

### Volume 190 / 2019

Acta Arithmetica 190 (2019), 49-56
MSC: Primary 12F12; Secondary 11R29.
DOI: 10.4064/aa170930-22-8
Published online: 14 June 2019

#### Abstract

We prove that for any finite solvable group $G$, there exist infinitely many cyclic extensions $K/\mathbb Q$ and Galois extensions $M/\mathbb Q$ such that the Galois group $\newcommand{\Gal}{\mathrm{Gal}}\Gal(M/K)$ is isomorphic to $G$ and $M/K$ is unramified. We can choose the base field $K$ having relatively small degree compared to our previous article [Osaka J. Math. 52 (2015)].