Absolute reconstruction of number fields from the Deligne-Ribet monoids

Following Cornelissen, Li, Marcolli, and Smit, this short paper proves that the field structure of a number field $K$ can be reconstructed from the pair $(DR_K, I_K)$ of the Deligne-Ribet monoid $DR_K$ and the submonoid $I_K$ of $DR_K$, when $K$ is the rational number field, or an imaginary quadratic field. The general-case reconstruction is also discussed, which is more abstract than the case of rational and imaginary quadratic fields.