# Math Help - Prove Z[sqrt(d)] is not a UFD when d=1 mod 4

1. ## Prove Z[sqrt(d)] is not a UFD when d=1 mod 4

Let $d \equiv 1 \pmod{4}$ be a square-free integer with $d \neq 1$. By considering the factorisation of $d-1$, show that the ring $\mathbb{Z}[\sqrt{d}]$ is never a Unique Factorisation Domain.

2. Originally Posted by Boysilver
Let $d \equiv 1 \pmod{4}$ be a square-free integer with $d \neq 1$. By considering the factorisation of $d-1$, show that the ring $\mathbb{Z}[\sqrt{d}]$ is never a Unique Factorisation Domain.
first show that 2 is irreducible: if $2=(m+n\sqrt{d})(r+s \sqrt{d}),$ then $4=(m^2-n^2d)(r^2-s^2d).$ but the equation $x^2-y^2d= \pm 2$ has no integer solution since $y^2d \pm 2 \equiv y^2 \pm 2 \equiv 2 \ \text{or} \ 3 \mod 4.$

now, if $\mathbb{Z}[\sqrt{d}]$ was a UFD, then, because 2 is irreducible, it'd have to be prime. therefore, since $2 \mid d-1 = (1 + \sqrt{d})(-1 + \sqrt{d})$, we'd have either $2 \mid 1 + \sqrt{d}$ or $2 \mid -1 + \sqrt{d},$ which is impossible.