Primes in denominators of algebraic numbers

TL;DR

This paper investigates the primes dividing denominators of algebraic numbers, characterizes their behavior in number fields, and relates these primes to class groups and algebraic properties of the numbers.

Contribution

It introduces a set X(K,γ) describing primes related to algebraic numbers and links its properties to class groups and principal ideal domains.

Findings

Characterization of primes in denominators of algebraic numbers.

Connection between X(K,γ) and class group generation.

Existence of a finite set S determining X(K,γ) across all number fields.

Abstract

Denote the set of algebraic numbers as $\overline{\mathbb{Q}}$ and the set of algebraic integers as $\overline{\mathbb{Z}}$. For $\gamma\in\overline{\mathbb{Q}}$, consider its irreducible polynomial in $\mathbb{Z}[x]$, $F_{\gamma}(x)=a_nx^n+\dots+a_0$. Denote $e(\gamma)=\gcd(a_{n},a_{n-1},\dots,a_1)$. Drungilas, Dubickas and Jankauskas show in a recent paper that $\mathbb{Z}[\gamma]\cap \mathbb{Q}=\{\alpha\in\mathbb{Q}\mid \{p\mid v_p(\alpha)<0\}\subseteq \{p\mid p|e(\gamma)\}\}$. Given a number field $K$ and $\gamma\in\overline{\mathbb{Q}}$, we show that there is a subset $X(K,\gamma)\subseteq \text{Spec}(\mathcal{O}_K)$, for which $\mathcal{O}_K[\gamma]\cap K=\{\alpha\in K\mid \{\mathfrak{p}\mid v_{\mathfrak{p}}(\alpha)<0\}\subseteq X(K,\gamma)\}$. We prove that $\mathcal{O}_K[\gamma]\cap K$ is a principal ideal domain if and only if the primes in $X(K,\gamma)$ generate the class group of $\mathcal{O}_K$. We show that given $\gamma\in \overline{\mathbb{Q}}$, we can find a finite set $S\subseteq \overline{\mathbb{Z}}$, such that for every number field $K$, we have $X(K,\gamma)=\{\mathfrak{p}\in\text{Spec}(\mathcal{O}_K)\mid \mathfrak{p}\cap S\neq \emptyset\}$. We study how this set $S$ relates to the ring $\overline{\mathbb{Z}}[\gamma]$ and the ideal $\mathfrak{D}_{\gamma}=\{a\in\overline{\mathbb{Z}}\mid a\gamma\in\overline{\mathbb{Z}}\}$ of $\overline{\mathbb{Z}}$. We also show that $\gamma_1,\gamma_2\in \overline{\mathbb{Q}}$ satisfy $\mathfrak{D}_{\gamma_1}=\mathfrak{D}_{\gamma_2}$ if and only if $X(K,\gamma_1)=X(K,\gamma_2)$ for all number fields $K$.