Orders of algebraic numbers in finite fields

TL;DR

This paper investigates the orders of algebraic numbers' reductions in finite fields, providing new results for degrees two and three, and exploring distribution and density properties under GRH.

Contribution

It extends understanding of algebraic number orders in finite fields, especially for degrees two and three, and introduces new density and distribution results under GRH.

Findings

Established order behavior for degree two algebraic numbers.

Proved positive lower density results for degree three reductions.

Presented an almost equidistribution result for linear recurrences modulo primes.

Abstract

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always almost maximal" assuming the generalized Riemann hypothesis, but unconditional results remain modest. We consider higher degree variants under GRH. First, we modify an argument of Roskam to settle the case where $\alpha$ and the reduction have degree two. Second, we give a positive lower density result when $\alpha$ is of degree three and the reduction is of degree two. Third, we give higher rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.