Character sums and primitive roots in algebraic number fields

Let ν denote a totally positive integer of an algebraic number fieldK such that ν is a least primitive root modulo a prime ideal \(\mathfrak{p}\) ofK, least in the sense that its normNν is minimal. One of the simplest questions that presents itself is that of the order of magnitude ofNν in comparison toN \(\mathfrak{p}\) . In the present paper the following bound is shown: $$N_\nu<< N\mathfrak{p}^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} + a} for fixed a > 0.$$ The proof of this result is based on deep estimates for certain character sums inK.