Show that the Quadratic Field $\Bbb Q(\sqrt{10})$ does not have the unique factorization property

Question

Directly show that $\Bbb Q(\sqrt{10})$ does not have the unique factorization property by considering the factorizations:

$$6 = 2\times 3 = (4 + \sqrt{10})\,(4 - \sqrt{10}).$$

Answer 1

You have to prove that these factorisations do not involve associated elements, i.e. there exists no unit $u\in\mathbf Q(\sqrt{10})$ such that $4+\sqrt{10}=2u$ or $3u$, and similarly for $4-\sqrt{10}$.

Hint:

Use the norm.