  For an algebraic number $\alpha$ of degree $n$, let $\mathcal{M}_{\alpha}$ be
the $\mathbb{Z}$-module generated by $1,\alpha ,\ldots ,\alpha^{n-1}$; then
$\mathbb{Z}_{\alpha}:=\{\xi\in\mathbb{Q} (\alpha ):\,
\xi\mathcal{M}_{\alpha}\subseteq\mathcal{M}_{\alpha}\}$ is the ring of scalars
of $\mathcal{M}_{\alpha}$. We call an order of the shape $\mathbb{Z}_{\alpha}$
\emph{rationally monogenic}. If $\alpha$ is an algebraic integer, then
$\mathbb{Z}_{\alpha}=\mathbb{Z}[\alpha ]$ is monogenic. Rationally monogenic
orders are special types of invariant orders of binary forms, which have been
studied intensively. If $\alpha ,\beta$ are two
$\text{GL}_2(\mathbb{Z})$-equivalent algebraic numbers, i.e., $\beta =(a\alpha
+b)/(c\alpha +d)$ for some
$\big(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\big)\in\text{GL}_2(\mathbb{Z})$,
then $\mathbb{Z}_{\alpha}=\mathbb{Z}_{\beta}$. Given an order $\mathcal{O}$ of
a number field, we call a $\text{GL}_2(\mathbb{Z})$-equivalence class of
$\alpha$ with $\mathbb{Z}_{\alpha}=\mathcal{O}$ a \emph{rational
monogenization} of $\mathcal{O}$.
  We prove the following. If $K$ is a quartic number field, then $K$ has only
finitely many orders with more than two rational monogenizations. This is best
possible. Further, if $K$ is a number field of degree $\geq 5$, the Galois
group of whose normal closure is $5$-transitive, then $K$ has only finitely
many orders with more than one rational monogenization. The proof uses
finiteness results for unit equations, which in turn were derived from
Schmidt's Subspace Theorem. We generalize the above results to rationally
monogenic orders over rings of $S$-integers of number fields. Our results
extend work of B\'{e}rczes, Gy\H{o}ry and the author from 2013 on multiply
monogenic orders.
