Variations on a theme of Schinzel and W\'ojcik

TL;DR

This paper investigates the conditions under which there are infinitely many primes with specific order relations of algebraic numbers modulo those primes, extending known results and exploring new families of such pairs in rational and quadratic fields.

Contribution

It introduces new classes of pairs of algebraic numbers for which the order relation holds infinitely often, including a result for all pairs of the form (A, 2), and extends the theorem to imaginary quadratic fields.

Findings

Proved infinitely many primes with greater order for certain pairs

Established that (A, 2) pairs satisfy the property for all positive integers A

Extended the theorem to algebraic integers in imaginary quadratic fields

Abstract

Schinzel and W\'ojcik have shown that if $\alpha, \beta$ are rational numbers not $0$ or $\pm 1$, then $\mathrm{ord}_p(\alpha)=\mathrm{ord}_p(\beta)$ for infinitely many primes $p$, where $\mathrm{ord}_p(\cdot)$ denotes the order in $\mathbb{F}_p^{\times}$. We begin by asking: When are there infinitely many primes $p$ with $\mathrm{ord}_p(\alpha) > \mathrm{ord}_p(\beta)$? We write down several families of pairs $\alpha,\beta$ for which we can prove this to be the case. In particular, we show this happens for "100\%" of pairs $A,2$, as $A$ runs through the positive integers. We end on a different note, proving a version of Schinzel and W\'{o}jcik's theorem for the integers of an imaginary quadratic field $K$: If $\alpha, \beta \in \mathcal{O}_K$ are nonzero and neither is a root of unity, then there are infinitely many maximal ideals $P$ of $\mathcal{O}_K$ for which $\mathrm{ord}_P(\alpha) = \mathrm{ord}_P(\beta)$.