Monogenic fields with odd class number Part II: even degree

TL;DR

This paper extends Gauss's classical result by proving the infinitude of certain monogenic fields with odd class number for even degrees and analyzes their class group structures, providing new average bounds for 2-torsion elements.

Contribution

It generalizes Gauss's theorem to higher even degrees and computes the first p-torsion averages for primes dividing the degree, advancing understanding of class group heuristics.

Findings

Infinitely many $S_n$-fields of even degree with odd class number exist.

Established bounds and exact averages for 2-torsion in class groups of monogenised fields.

First p-torsion averages computed for primes dividing the degree, informing class group heuristics.

Abstract

In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisely conditional on a tail estimate) using a parametrisation of Wood. These averages are the first $p$-torsion averages to be calculated for $p$ not coprime to the degree (in degree at least $3$), shedding light on the question of Cohen-Lenstra-Martinet-Malle type heuristics for class groups and narrow class groups at "bad" primes.