Orders of algebraic numbers in finite fields

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.