Ideal class group of $ \Bbb{Z}[ \sqrt{-5}]$

Question

Ideal class group of $ \Bbb{Z}[ \sqrt{-5}]$ is known to be {$(1),I=(2,1+\sqrt{-5}$}. This is a group of order $2$, so $I^2$ must be $(1)$. But after calculating,$I^2=(2)$. $(1)$ is not the same ideal as $(2)$. Where am I mistaken ?

Answer 1

The class group is fractional ideals mod principal ideals. $(2)$ is a principal ideal so it's equivalent to $(1)$ in the class group.