On Artin's conjecture

Abstract

Let F be a family of number fields which are normal and of finite degree over a given number field K. Consider the lattice L(scF) spanned by all the elements of F . The generalized Artin problem is to determine the set of prime ideals of K which do not split completely in any element H of L(scF), H≠ K. Assuming the generalized Riemann hypothesis and some mild restrictions on F , we solve this problem by giving an asymptotic formula for the number of such prime ideals below a given norm. The classical Artin conjecture on primitive roots appears as a special case. In another case, if F is the family of fields obtained by adjoining to Q the q-division points of an elliptic curve E over Q , the Artin problem determines how often E( F p ) is cyclic. If E has complex multiplication, the generalized Riemann hypothesis can be removed by using the analogue of the Bombieri-Vinogradov prime number theorem for number fields.