Caliber numbers of real quadratic fields

TL;DR

This paper establishes lower bounds for the caliber number of real quadratic fields using splitting primes, classifies fields with small caliber numbers, and does so without relying on assumptions about the Dedekind zeta function.

Contribution

It provides a new method to bound and classify real quadratic fields by their caliber number based on splitting primes, independent of zeta function assumptions.

Findings

Identified all real quadratic fields with caliber number 1.

Identified all real quadratic fields with caliber number 2 (when d ≠ 5 mod 8).

Established lower bounds for the caliber number using splitting primes.

Abstract

We obtain lower bound of caliber number of real quadratic field $K=\FQ(\sqrt{d})$ using splitting primes in $K$. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if $d$ is not 5 modulo 8. In both cases, we don't rely on the assumption on $\zeta_K(1/2)$.