Leírás
Online Number Theory Seminar
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 $\mathbb{Z}_{\alpha}$ \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 polynomials, which were introduced by Birch and Merriman (1972), Nakagawa (1989), and Simon (2001).
If $\alpha ,\beta$ are two $\text{GL}_2(\mathbb{Z})$-equivalent algebraic numbers, i.e., $\beta =(a\alpha +b)/(c\alpha +d)$ for some
$\Big(\begin{matrix}a&b\\c&d\end{matrix}\Big)\in\text{GL}_2(\mathbb{Z})$, 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 $\mathcal{O}$.
It is known that every order of a number field has at most finitely many rational monogenizations. Among other things, we discuss our new result that if $K$ is a number field of degree $n\geq 5$
with normal closure having maximal Galois group $S_n$, then apart from at most finitely many exceptions, every order of $K$ has at most one rational monogenization.
For access please contact the organizers (ntrg[at]science.unideb.hu).