For question #1, see this sequence, which contains all values $c$ such that $\mathbb{Z}[c]$ \mathbb{Z}[\sqrt{c}]$is Euclidean: http://oeis.org/A048981 Note that these are the only such quadratic fields which are Euclidean. For question #3, it is not solved completely since it is not known which values of$c>0$produce a unique factorization domain. However, the problem is solved for$c<0$completely via the Stark-Heegner theorem. The problem in general dates back to Gauss, and is known as the class number problem. Wikipedia has good info. Using the answers to these, we can answer #2 since there are only a finite number of Euclidean quadratic fields. 3 added 180 characters in body For question #1, see this sequence, which contains all values$c$such that$\mathbb{Z}[c]$is Euclidean: http://oeis.org/A048981 Note that these are the only such quadratic fields which are Euclidean. For question #3, it is not solved completely since it is not known which values of$c>0$produce a unique factorization domain. However, the problem is solved for$c<0$completely via the Stark-Heegner theorem. The problem in general dates back to Gauss, and is known as the class number problem. Wikipedia has good info. Using the answers to these, we can answer #2 since there are only a finite number of Euclidean quadratic fields. 2 more info; added 116 characters in body For question #1, see this sequence, which contains all values$c$such that$\mathbb{Z}[c]$is Euclidean: http://oeis.org/A048981 Note that these are the only such quadratic fields which are Euclidean. For question #3, it is not solved completely since it is not known which values of$c>0$produce a unique factorization domain. However, the problem is solved for$c<0\$ completely via the Stark-Heegner theorem.