On a certain invariant for real quadratic fields

Abstract

Let $K = \mathbf{Q}(\sqrt{m})$ be a real quadratic field, $\mathcal{O}_K$ its ring of integers and $G = \operatorname{Gal}(K/\mathbf{Q})$. For $\gamma \in H^1(G, \mathcal{O}_K^{\times})$, we associate a module $M_c/P_c$ for $\gamma = [c]$. It is known that $M_c/P_c \approx \mathbf{Z}/\Delta_m \mathbf{Z}$ where $\Delta_m = 1$ or 2 and we will determine $\Delta_m$.