Preassigned character values in the Gaussian integers

Abstract

Let γ 1, γ 2,…, γ t be distinct prime Gaussian integers. With k an odd positive rational integer, take ω 1, ω 2,…, ω t to be k-th roots of unity, not necessarily distinct nor primitive. We show there exist infinitely many prime Gaussian integers π having a k-th-power character χ such that χ( γ i ) = ω i , 1 ≤ i ≤ t.