On Artin's conjecture |
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Authors: | MRam Murty |
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Institution: | School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 USA |
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Abstract: | Let 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 . 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 , 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 is the family of fields obtained by adjoining to the q-division points of an elliptic curve E over , the Artin problem determines how often E(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. |
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