Infinite class field tower with small root discriminant

Zugan Xing, Qi Liu

Published 2024-06-02Version 1

We extend Schoof's theorem from cyclic case to any finite case and apply this to construct a class of $\mathbb Z/m\mathbb Z \rtimes \mathbb Z/\varphi(n)\mathbb Z$ extensions of $\mathbb Q$, where $m$ is either a power of $2$ or an odd integer, and $n$ be any integer such that $m$ divides $n$. As an application, we give some number fields with small root discriminant, having infinite $p$-class field tower when $p=3, 5, 7$.