Thread: For what NEGATIVE values of d is Z[sqrt[d]] a Euclidean domain?

I was asked to prove that the ring Z[sqrt[d]] is not euclidean for any d <= -3. (Note - The resulting ring it not contained in R!)

I proved it for certain d's easily (For example d=-3), by showing that it is not a UFD, but am not sure how to prove it in the more general case. (Or whether Z[sqrt[d]] is a UFD for some d's although it is not Euclidean)

Could anyone suggest a solution or intuition as to why is it not a Euclidean Domain?

for $\displaystyle Z[\sqrt{-2}]$ note $\displaystyle (1-\sqrt{-2})(1+\sqrt{-2}) = 3 $ If this were a euclidean domain, it would be a PID, which means primes are irreducible. Since neither of the factors are units, its safe to say this is not a Euclidean domain

Same reasoning for $\displaystyle Z[\sqrt{-1}$ take $\displaystyle (1-\sqrt{-1})(1+\sqrt{-1}) = 2$ with neither factors being units.