On S3-extensions with infinite class field tower

TL;DR

This paper constructs specific $S_3$-extensions of the rational numbers with infinite 3-class field towers, demonstrating minimal ramification and providing examples with the smallest known root discriminant among such fields.

Contribution

It introduces new $S_3$-extensions with infinite 3-class field towers and minimal ramification, advancing understanding of class field towers with controlled ramification.

Findings

Constructed $S_3$-extensions with infinite 3-class field towers

Achieved minimal ramification with only three primes ramified

Identified examples with smallest known root discriminant among such fields

Abstract

We construct a class of $S_3$-extensions of $\Q$ with infinite 3-class field tower in which only three primes ramify. As an application, we obtain an $S_3$-extension of $\Q$ with infinite 3-class field tower with smallest known (to the author) root discriminant among all fields with infinite 3-class field tower.