Orders with few rational monogenizations
Volume 210 / 2023
Abstract
Recall that a monogenic order is an order of the shape $\mathbb {Z}[\alpha ]$, where $\alpha $ is an algebraic integer. This is generalized to orders $\mathbb {Z}_{\alpha }$ for not necessarily integral algebraic numbers $\alpha $ as follows. 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 $\mathbb {Z}_{\alpha }$ rationally monogenic. If $\alpha $ is an algebraic integer, then $\mathbb {Z}_{\alpha }=\mathbb {Z} [\alpha ]$ is monogenic. In fact, rationally monogenic orders are special cases of invariant rings of polynomials or binary forms, which were introduced by Birch and Merriman (1972), Nakagawa (1989), and Simon (2001). If $\alpha ,\beta $ are two ${\rm GL}_2(\mathbb {Z} )$-equivalent algebraic numbers, i.e., $\beta =\frac {a\alpha +b}{c\alpha +d}$ for some $\big (\begin {smallmatrix}a&b\\c&d\end {smallmatrix}\big )\in {\rm GL}_2(\mathbb {Z} )$, then $\mathbb {Z}_{\alpha }=\mathbb {Z}_{\beta }$. Given an order $\mathcal {O}$ of a number field, we call a ${\rm GL}_2(\mathbb {Z} )$-equivalence class of $\alpha $ with $\mathbb {Z}_{\alpha }=\mathcal {O}$ a 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 Bérczes, Győry and the author from 2013 on multiply monogenic orders.
