Hatcher-Topology-of-Numbers · Problem 8.3.11

The key to this problem is understanding how the class group structure of an order $R_\Delta$ in an imaginary quadratic field determines ideal classes. You'll need to factor $\Delta$ and compute which ideals are principal versus genuinely non-principal, then group them by their equivalence under multiplication by principal ideals. Strict equivalence further refines this by distinguishing ideals that differ by units or conjugation. The class number tells you how many equivalence classes exist, and explicit computation of reduced forms reveals their representatives.

The key to this problem is understanding how the class group structure of an order $R_\Delta$ in an imaginary quadratic field determines ideal classes. You'll need to factor $\Delta$ and compute which ideals are principal versus genuinely non-principal, then group them by their equivalence under multiplication by principal ideals. Strict equivalence further refines this by distinguishing ideals that differ by units or conjugation. The class number tells you how many equivalence classes exist, and explicit computation of reduced forms reveals their representatives.