I know the very basic basics of $\mathbb{Z}[\sqrt{14}]$. Numbers in it are of the form $a + b \sqrt{14}$, with $a, b \in \mathbb{Z}$. Numbers like $-3 + \sqrt{14}$ and $7 - 8 \sqrt{14}$. The norm function is $N(a + b \sqrt{14}) = a^2 - 14b^2$, which I'm told is not an Euclidean function even after the absolute value adjustment.

Proposition $4.11$ in this paper by Franz Lemmermeyer

<http://www.rzuser.uni-heidelberg.de/~hb3/publ/survey.pdf>

mentions $\mathbb{Z}\left[\sqrt{14}, \frac{1}{2}\right]$. I don't think I've ever read about a domain like that before, except perhaps in a very general way that the specifics eluded me.

I'm guessing $\mathbb{Z}\left[\sqrt{14}, \frac{1}{2}\right]$ contains all the same numbers of $\mathbb{Z}[\sqrt{14}]$ as well as some other numbers. **What's the form of those other numbers? What are some concrete examples of those other numbers?**

P.S. Proposition $4.11$ is on page $14$ of $56$ of the PDF. Looks like a brilliant survey. I need to print it out and sit down to read it beginning to end.