Orders with few rational monogenizations

Abstract

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.