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 \}.

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