Caliber numbers of real quadratic fields

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 $\zeta_K(1/2)$.