Science-advisor
REGISTER info/FAQ
Login
username
password
     
forgot password?
register here
 
Research articles
  search articles
  reviews guidelines
  reviews
  articles index
My Pages
my alerts
  my messages
  my reviews
  my favorites
 
 
Stat
Members: 3667
Articles: 2'599'751
Articles rated: 2609

07 February 2025
 
  » arxiv » 2301.01552

 Article overview



Orders with few rational monogenizations
Jan-Hendrik Evertse ;
Date 4 Jan 2023
AbstractTo an algebraic integer $alpha$, we can attach the order $mathbb{Z} [alpha ]$, and an order of this shape is called emph{monogenic}. For a not necessarily integral algebraic number $alpha$, say of degree $n$, we define an order $mathbb{Z}_{alpha}$ as follows: let $mathcal{M}_{alpha}$ be the $mathbb{Z}$-module generated by $1,alpha ,ldots ,alpha^{n-1}$; then $mathbb{Z}_{alpha}:={xiinmathbb{Q} (alpha ):, ximathcal{M}_{alpha}subseteqmathcal{M}_{alpha}}$ is the ring of scalars of $mathcal{M}_{alpha}$. We call an order of the shape $mathbb{Z}_{alpha}$ rationally monogenic. In fact, rationally monogenic orders are special types of invariant orders of binary forms, and these have been intensively studied. If $alpha ,eta$ are two $ ext{GL}_2(mathbb{Z})$-equivalent algebraic numbers, i.e., $eta =(aalpha +b)/(calpha +d)$ for some $ig(egin{smallmatrix}a&b\c&dend{smallmatrix}ig)in ext{GL}_2(mathbb{Z})$, then $mathbb{Z}_{alpha}=mathbb{Z}_{eta}$. Given an order $mathcal{O}$ of a number field, we call any $ ext{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 five times 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.
Source arXiv, 2301.01552
Services Forum | Review | PDF | Favorites   
 
Visitor rating: did you like this article? no 1   2   3   4   5   yes

No review found.
 Did you like this article?

This article or document is ...
important:
of broad interest:
readable:
new:
correct:
Global appreciation:

  Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.






ScienXe.org
» my Online CV
» Free

home  |  contact  |  terms of use  |  sitemap
Copyright © 2005-2025 - Scimetrica