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In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are aprimitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967,Hooley showed that this conjecture follows from the Generalized RiemannHypothesis (GRH). In this paper we consider variants of the primitive rootconjecture for number fields and for matrices. All results are conditional onGRH.  For an algebraic number field $K$ and some element $\alpha \in K$, we examinethe order of $\alpha$ modulo various rational primes $p$. We extend previousresults of Roskam which only worked for quadratic extensions $K/\mathbb{Q}$ tomore general field extensions of higher degree. This allows us to show that forspecial number fields $K/\mathbb{Q}$ whose Galois group satisfies some furtherconditions, $\alpha$ is of almost maximal order mod $p$ for almost all rationalprimes $p$ which factor into primes of degree 2 in $K$.  We also consider Artin's primitive root conjecture for matrices. Given amatrix $A\in\text{GL}_n(\mathbb{Q})$, we examine the order of $A\bmod p$ in$\text{GL}_n(\mathbb{F}_p)$ for various primes $p$, which turns out to beequivalent to the number field setting.

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