A Note on Distribution of Primitive Roots in a Finite Field

Abstract:

      Let f₁(x) , …,f_r(x) denote quadratfree polynomials over GF(q) that are relatively prime in pairs and of degree≥1. We have proved that if q≥(k-1)²(2^rω(q-1)-1)², then all the f_i(ξ) are primitive roots of GF(q) for some ξ∈GF(q), where k =degf₁ … degf_r . With extending the results obtained by Wang Juping, Sun Qi and Han Wenbao respectively, we have also given a quantitative analysis of Carlitz's result.