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.

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