Primitive roots in quadratic fields, II Original Research Article

We consider an analogue of Artinʹs primitive root conjecture for algebraic numbers which are not units in quadratic fields. Given such an algebraic number α, for a rational prime p which is inert in the field, the maximal possible order of α modulo (p) is p2−1. An extension of Artinʹs conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that for any choice of 113 algebraic numbers satisfying a certain simple restriction, at least one of the algebraic numbers has order at least image for infinitely many inert primes p.