On the least primitive root in number fields

Abstract

Let K be an algebraic number field and \( \mathfrak{O} \)K its ring of integers. For any prime ideal \( \mathfrak{p} \), the group \( (\mathfrak{O}_K /\mathfrak{p})* \) of the reduced residue classes of integers is cyclic. We call any element of a generator of the group \( (\mathfrak{O}_K /\mathfrak{p})* \) a primitive root modulo \( \mathfrak{p} \). Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal \( \mathfrak{p} \) under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.