Is there a general procedure to check whether or not a prime ideal of the ring of integers $O_K$ is principal. In my case $K$ is a quadratic field, i.e $\mathbb{Q}(\sqrt {d})$, with $d$ square-free.

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    $\begingroup$ One way to see it isn't is by looking at its norm and seeing that no element of $\mathcal O_K$ has that norm. $\endgroup$ – Thomas Poguntke May 9 '15 at 16:57
  • $\begingroup$ If $O_K$ is a unique factorization domain, things are very simple. It's when it lacks unique factorization that this question becomes much more interesting. Let's take $d = -5$ for example. Does $\langle 1 + \sqrt{-5}, 2 \rangle$ look like a prime ideal? Does it look like a principal ideal? $\endgroup$ – David R. May 9 '15 at 21:41
  • $\begingroup$ The ideal looks like it may be prime, but I'm sure how I can be certain of that. Also it's not clear to me on how to show the ideal you mention is principal or not. As you may know, $\langle 2,\sqrt{11}+1\rangle=\langle \sqrt{11}+3\rangle$ in the field $O_K$, with $K=\mathbb{Q}(\sqrt{11})$. $\endgroup$ – Andrew Brick May 10 '15 at 6:15

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