## Exhibit an extended principal generator for an ideal in an algebraic integer ring

Let $A = (3,1+2\sqrt{-5})$ be an ideal in the ring of integers of $K = \mathbb{Q}(\sqrt{-5})$. Find an algebraic integer $\kappa$ such that $A = (\kappa) \cap \mathcal{O}_E$, where $E = K(\kappa)$.

We saw in the text that the class number of $K$ is 2. Note that $A^2 = (3)$, as indeed $3 = (1+2\sqrt{-5})^2 - 2 \cdot 3^2$. Let $\kappa = \sqrt{3}$. By our proof of Theorem 10.6, $A = \{ \tau \in \mathcal{O}_K \ |\ \tau/\kappa \in \mathcal{O}_K \}$.