Factoring an integer over quadratic ring

Question

The discriminant of the ring $\mathbb{Z}[\sqrt{10}]$ is 40. Since 2 divides 40, 2 is ramified in this ring. I know that the ideal $(2)$ factors as $(2,\sqrt{10})^2$, but I cannot find a factorization of 2 in this ring. The equation $x^2-10y^2=\pm 2$ has no solutions. How would I find a factorization of 2 in this ring (or is there none)? I would like a method that can be generalized to other quadratic integer rings.

Answer 1

I assume that you mean a factorization of $2$ into irreducible elements in this ring, then note that the fact that $x^2-10y^2= \pm 2$ has no solutions, i.e. there are no elements of norm $2$ in $\Bbb Z[\sqrt{10}]$ implies that $2$ is an irreducible element in $\Bbb Z[\sqrt{10}]$, since if we ahve $2=ab$, then taking norms gives $4=N(2)=N(a)N(b)$, but since there are no elements of norm $\pm 2$, we get that $N(a)=\pm 1$ or $N(b) = \pm 1$, so either $a$ or $b$ is a unit.