Jacobi sums and irreducible polynomials with prescribedtrace and restricted norm

摘要

Let Ψ be any multiplicative character of the finite field F_p~vofp~velements, and define the special Jacobi sum J_t(ψ)= Σx_i∈F_p~v,x_1+…+t=1ψ(1...t). Here we show how to express J_t(ψ) as a ratio of Eisenstein sums involvinga power or lift of ψ.Using known evaluations for Eisenstein sums we give acomplete determination of J_t(ψ)when the ordersof ψdivides 8 or 12. Theseresults are then applied to the problem to determine the number of irreduciblepolynomials of fixed degree over any finite field F_qwith prescribed trace, and norm lying in a specified s-power coset of F_q~*,wheresdivides 8 or 12. Wenote that this determination may be extended to the case s = 24 in similarfashion, though not without additional complications.