Question

Explain why $\mathbf{\mathbb{Z}}\mathbf{\left[}\sqrt{\mathbf{-}\mathbf{5}}\mathbf{\right]}$ is not a Euclidean domain for any function $\mathit{\delta }$.

Short answer

$\mathbb{Z}\left[\sqrt{-5}\right]$is not a Euclidean domain because it is not a unique factorization domain.

Step-by-step solution

Let ℤ[-5]  is a Euclidean domain

It is known that every Euclidean domain is also a unique factorization domain (UFD).

So, if$\mathbb{Z}\left[\sqrt{-5}\right]$is a Euclidean domain then it must be a UFD as well.

Prove that ℤ[-5] is not a UFD

It has been proved in the texts as well that $\mathrm{\mathbb{Z}}\left[\sqrt{-5}\right]$is not a UFD as the element 6 in $\mathrm{\mathbb{Z}}\left[\sqrt{-5}\right]$has two factorizations:

$6=2\cdot 3$and $6=(1+\sqrt{-5})(1-\sqrt{-5})$

Since the element 6 has two factorizations in$\mathrm{\mathbb{Z}}\left[\sqrt{-5}\right]$,therefore it is not a UFD.

Then it is not a Euclidean domain as well.

Conclusion

Thus, it can be concluded that$\mathrm{\mathbb{Z}}\left[\sqrt{-5}\right]$is not a Euclidean domain.