Let $c$ be an integer, not necessarily positive and $|c|$ not a square. Let $\mathbb{Z}[\sqrt{c}]$ be the set of complex numbers $$a+b\sqrt{c}, \quad a, b\in \mathbb{Z},$$ which form a subring of the ring $\mathbb{C}$ under the usual addition and multiplication.

Are the following questions completely solved?

For what $c$ is $\mathbb{Z}[\sqrt{c}]$ a Euclidean domain?

For what $c$ is $\mathbb{Z}[\sqrt{c}]$ a UFD (unique factorization domain) but not Euclidean ?

For what $c$ is $\mathbb{Z}[\sqrt{c}]$ not a UFD ?

I know that for $c=-1$, Question 1 has a positive answer; for $c=-5$, Question 3 has a positive answer.

reallymean to be writing about the ring Z[sqrt(c)] or do you want to write about the ring of integers of Q(sqrt(c))? $\endgroup$ – KConrad Jan 10 '11 at 19:32