## Verify an ideal factorization

Verify the following two equalities of ideals in the ring of integers in $\mathbb{Q}(\sqrt{-5})$: $(3) = (3,1+2\sqrt{-5})(3,1-2\sqrt{-5})$ and $(7) = (7,1+2\sqrt{-5})(7,1-2\sqrt{-5})$.

Note that $(3,1+2\sqrt{-5})(3,1-2\sqrt{-5}) = (9, 3-6\sqrt{-5}, 3+6\sqrt{-5}, 21) \subseteq (3)$. Moreover, $3 = 3 \cdot 3 - 3(1-2\sqrt{-5}) - 3(1+2\sqrt{-5}) \in (3,1+2\sqrt{-5})(3,1-2\sqrt{-5})$.

Note that $(7,1+2\sqrt{-5})(7,1-2\sqrt{-5}) = (49, 7-14\sqrt{-5}, 7+14\sqrt{-5}, 21) \subseteq (7)$. Moreover, $7 = 7 \cdot 7 - 21(1-2\sqrt{-5}) - 21(1+2\sqrt{-5}) \in (3,1+2\sqrt{-5})(3,1-2\sqrt{-5})$.