What are the reduced elements of $\mathbb Q(\sqrt{30})$? 
What are the reduced elements of $\mathbb Q(\sqrt{30})$ ?

From the definition here(on page $32$);
An element $\beta\in\mathbb Q(\sqrt{d})$ is said to be reduced, if $\beta>1$ and $-1<\beta'<0$ and $\beta'$ is the image of $\beta$ under the nontrivial automorphism which sends $\sqrt{d}$ to $-\sqrt{d}$ 
If $\beta$ is of the form $a+b\sqrt{30}$, then $0>\beta'=a-b\sqrt{30}>-1$ 
this looks too complicated to determine all reduced elements
So if $\beta$ is reduced, then clearly it is irrational, and we've a theorem which states: For $\beta\in\mathbb Q(\sqrt{d})\setminus\mathbb Q$, there are only finitely many reduced elements for any $D\in\mathbb N_{>0}$, where $D$ is the discriminant $(B^2-4AC)$ of the minimal polynomial $(AT^2+BT+C=0$ and $ \gcd(A,B,C)=1)$ of $\beta$. Is this useful ?
For example if we define $\beta_0=\sqrt{30}+\lfloor\sqrt{30}\rfloor,\quad\beta_{n+1}=\frac{1}{\beta_n-\lfloor\beta_n\rfloor}$ than $(\beta_n)_n$ is a periodic sequence of reduced elements of $\mathbb Q(\sqrt{30})$, but these are not all.
Do you have an idea ?

For any $D>0$ there are only finitely many reduced elements $\beta$ whose minimal polynomial has discriminant $D$
The proof relies on the fact that there are only finitely many possible values for the coefficients $A,B$ in the minimal polynomial $A x^2+B x+C$ of $\beta$
Apply this result to $\mathbb{Q}\left(\sqrt{30}\right)$
Now $D=30m^2$ for some $m\in \mathbb{Z}$. 
Choose $A,B$ and calculate $\beta =\frac{-B+\sqrt{D}}{2A}$
