Decide whether a given linear congruence over an algebraic integer ring has a solution

Consider A = (3\sqrt{-5}, 10+\sqrt{-5}) as an ideal in \mathcal{O} = \mathbb{Z}[\sqrt{-5}]. Does 3\xi \equiv 5 mod A have a solution in \mathcal{O}?


Let D = ((3),A) = (3,1+\sqrt{-5}). By Theorem 9.13, our congruence has a solution if and only if 5 \equiv 0 mod D. But as we saw in a previous exercise, 5 \equiv 2 \not\equiv 0 mod D; so this congruence does not have a solution in \mathcal{O}.

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