TITLE: What does a number look like that's in $\mathbb{Z}\left[\sqrt{14}, \frac{1}{2}\right]$ that's not in $\mathbb{Z}[\sqrt{14}]$?
QUESTION [6 upvotes]: 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.

REPLY [6 votes]: If $a,b \in \mathbb R$, then $\mathbb{Z}[a,b]$ is the smallest ring that contains $a,b$; it is the same as the set of all polynomial expressions in $a,b$ with coefficients in $\mathbb{Z}$.
When $a=\sqrt{14}$ and $b=1/2$, all powers of $a$ reduce to an integer or to an integer times $a$. There is no reduction for powers of $b$, but all fractions can be reduced to the same denominator.
Therefore, a typical element of $\mathbb{Z}\left[\sqrt{14}, \frac{1}{2}\right]$ is of the form $\dfrac{u+v\sqrt{14}}{2^n}$, for $u,v \in \mathbb Z$ and $n \in \mathbb N$.