## ZZ[sqrt(-5)] is not norm-Euclidean

Show (via a counterexample) that $\mathbb{Z}[\sqrt{\text{-}5}]$ is not a Euclidean domain via the norm $N(a+b\sqrt{\text{-}5}) = a^2 + 5b^2$.

Suppose to the contrary that there exist $\pi = a_1 + a_2\sqrt{\text{-}5}$ and $\rho = b_1 + b_2\sqrt{\text{-}5}$ such that $1+\sqrt{\text{-}5} = 2\pi + \rho$ and $N(\rho) < N(2)$. Expanding this out and comparing real and imaginary parts yields the two equations $2a_1 + b_1 = 1$ and $2a_2 + b_2 = 1$ with $b_1^2 + 5b_2^2 < 4$. Now $b_2 = 0$, so that $2a_2 = 1$. But this equation has no solutions in the integers, and we have a contradiction.

So $\mathbb{Z}[\sqrt{\text{-}5}]$ is not Euclidean with respect to the norm $N(a+b\sqrt{\text{-}5}) = a^2 + 5b^2$.