Nonprincipal ideal in the integers adjoined a square root of minus five
= Nonprincipal ideal in the integers adjoined a square root of minus five
In $R=\mathbb Z[\sqrt{-5}]$, the ideal
$$
I=(2,1+\sqrt{-5})
$$
has additive index two. If $I=(a+b\sqrt{-5})$ were principal, multiplication by its generator would give
$$
[R:I]=|a^2+5b^2|=2,
$$
which has no solution in integers. Thus $I$ is not principal.