## Show that two ideals are distinct

Let $\mathcal{O}$ be the ring of integers in $\mathbb{Q}(\sqrt{-5})$. Show that the ideals $(3,1+2\sqrt{-5})$ and $(7,1-2\sqrt{-5})$ are distinct.

Suppose to the contrary that these ideals are equal. Then in particular, we have $3 = 7(a+b\sqrt{-5}) + (1-2\sqrt{-5})(c+d\sqrt{-5})$. Comparing coefficients yields the two integer equations $7a+c+10d = 0$ and $7b+d-2c = 0$. Mod 7, we have $c+3d\equiv 3$ and $d+5c \equiv 0$. Solving this system by substitution yields $0 \equiv 3$ mod 7; hence the system has no solution.

So these ideals are distinct.