A multiquadratic field generalization of Artin's conjecture (Q311459)

Among the conjectures due to Artin there is one that states that for any nonsquare integer \(a \neq -1\) the set of primes for which \(a\) is a primitive root modulo \(p\) has positive density. The present article deals with the following more general question: For which number fields \(K\) is the index of the unit group in \(({\mathcal O}_K/p{\mathcal O}_K)^\times\) equal to \(\frac{p-1}2\) for infinitely many \(p\)? It is easy to see that the index is at least \(p-1\) if each unit has norm \(+1\).NEWLINENEWLINEThe main result proved in this article is the following theorem: Let \(K\) be a totally real multiquadratic number field with unit grooup \({\mathcal O}_K^\times\), and assume that \(K\) contains a unit with norm \(-1\). Then the generalized Riemann Hypothesis implies that the primes that split in \(K\) and for which the image of \({\mathcal O}_K^\times\) in \(({\mathcal O}_K/p{\mathcal O}_K)^\times\) has index \((p-1)/2\) has a positive density, which can be given explicitly.