Question

Show that $\mathbf{\mathbb{Z}}\mathbf{\left[}\sqrt{\mathbf{-}\mathbf{6}}\mathbf{\right]}$ is not a UFD.

Short answer

It has been proved that$\mathrm{\mathbb{Z}}\left[\sqrt{-6}\right]$ is not a UFD.

Step-by-step solution

Factor 10 in 2 different ways

Considering the hint, factor 10 in two different ways.

Note that:

$\begin{array}{l}10=2\cdot 5\\ =(2+\sqrt{-6})\cdot (2-\sqrt{-6})\end{array}$

These factors are irreducible as there is no such a such that $N\left(a\right)=2$or 5.

Hence, 10 can be factorized in two different ways in$\mathrm{\mathbb{Z}}\left[\sqrt{-6}\right]$ .

Conclusion

Therefore, it can be concluded that$\mathrm{\mathbb{Z}}\left[\sqrt{-6}\right]$ is not a UFD.