Question

Show that $\mathbf{\mathbb{Z}}\mathbf{\left[}\sqrt{\mathbf{10}}\mathbf{\right]}$ is not a UFD.

Short answer

It has been proved that$\mathrm{\mathbb{Z}}\left[\sqrt{10}\right]$ is not a UFD.

Step-by-step solution

Factor 6 in 2 different ways

Considering the hint, factor 6 in two different ways.

Note that:

$\begin{array}{c}6=2\cdot 3\\ =(2+\sqrt{10})\cdot (-2+\sqrt{10})\end{array}$

These factors are irreducible as there is no integral solution of $x^2-10y^2=2,3$

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

Conclusion

Therefore, it can be concluded that $\mathrm{\mathbb{Z}}\left[\sqrt{10}\right]$is not a UFD.