## 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 $\mathcal{O}$.