Infinite class field tower with small root discriminant

TL;DR

This paper generalizes Schoof's theorem to construct infinite class field towers over cyclotomic fields, producing number fields with small root discriminant and infinite p-class field towers for p=3, 5, 7.

Contribution

It extends Schoof's theorem and applies it to generate new examples of number fields with infinite class field towers and small root discriminant.

Findings

Constructed infinite class field towers over cyclotomic fields.

Identified number fields with small root discriminant and infinite p-class towers for p=3, 5, 7.

Generalized Schoof's theorem to broader class of extensions.

Abstract

We generalize Schoof's theorem in 1986 and apply this to construct a class of Kummer extensions of the cyclotomic fields with infinite class tower. As an application, we give some number fields with a small root discriminant, which has an infinite $p$-class field tower when $p=3, 5, 7$.