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.

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