Artin's Primitive Root Conjecture in Number Fields and For Matrices (2306.15973v3)

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.