Artin's primitive root conjecture for quadratic fields
Artin's conjecture on primitive roots says that the set of primes for which a given integer \(a\) is a primitive root has a natural density, which is positive except when \(a = \pm 1\). There are two ways in which this conjecture can be generalized to arbitrary number fields: the obvious one, where the set of primes is replaced by the set of prime ideals, was proved under the assumption of GRH by \textit{G. Cooke} and \textit{P. Weinberger} [Commun. Algebra 3, 481-524 (1975; Zbl 0315.12001)]. The conjecture studied in this article is the following: let \(O\) denote the ring of integers of the number field \(K\), and consider the rational primes for which the order of a given \(\alpha \in K^\times\) inside the group \(C_p = (O/pO)^\times\) is equal to the exponent of \(C_p\); then this set has a natural density inside the set of all rational primes. The author proves this conjecture for quadratic number fields again assuming the truth of GRH and explains why, at least at present, this proof cannot be generalized to fields of degree \(> 2\).
scientific article; zbMATH DE number 1943341