0

From Artin algebra books. chapter 13 Quadratic Number field

For which negative integer $d\equiv 2\mod4 $ is the rings of integer in $\mathbb{Q}(\sqrt d)$ a unique factorization domain.

My works : i know that integer R in $\mathbb{Q}(\sqrt d)$are of the form $a +b \sqrt d$ for $a,b \in \mathbb{Z}$

i thinks $d= -6$ or may be $d=0$

Am i right/wrong ?

Any hints/solution will be appreciated .

thanks u

jasmine
  • 14,457

1 Answers1

2

The ring of integers in the imaginary quadratic number field $\Bbb{Q}(\sqrt{d})$, for $d<0$ squarefree, is a UFD if and only if $$d=-1,-2,-3,-7-,11,-19,-43,-67,-167,$$ by Baker and Stark (1967). For $d\equiv 2 \bmod 4$ we obtain $d=-2$. Note that $d=0$ does not satisfy the congruence.

For the proof, you do not need the result of Baker and Stark, but it is interesting anyway in this context.

Reference for the proof: This duplicate in case $d\not\equiv 1\bmod 4$.

Dietrich Burde
  • 130,978