Artin's Primitive Root Conjecture in Number Fields and For Matrices

TL;DR

This paper extends Artin's primitive root conjecture to algebraic number fields and matrices, showing under GRH that certain elements and matrices have near-maximal order modulo many primes, generalizing previous quadratic results.

Contribution

It generalizes primitive root conjecture results from quadratic fields to higher degree fields and matrices, under GRH, with specific Galois group and element constraints.

Findings

Almost maximal order of elements in number fields for many primes

Extension of quadratic case to higher degree fields

Equivalence between matrix orders and number field settings

Abstract

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH). In this paper we consider variants of the primitive root conjecture for number fields and for matrices. All results are conditional on GRH. For an algebraic number field $K$ and some element $\alpha \in K$, we examine the order of $\alpha$ modulo various rational primes $p$. We extend previous results of Roskam which only worked for quadratic extensions $K/\mathbb{Q}$ to more general field extensions of higher degree. Specifically, under some constraints on the Galois group of $K/\mathbb{Q}$ and on the element $\alpha\in K$, we show that $\alpha$ is of almost maximal order mod $p$ for almost all rational primes $p$ which factor into primes of degree 2 in $K$. We also consider Artin's primitive root conjecture for matrices. Given a matrix $A\in\text{GL}_n(\mathbb{Q})$, we examine the order of $A\bmod p$ in $\text{GL}_n(\mathbb{F}_p)$ for various primes $p$, which turns out to be equivalent to the number field setting.