Explicit lifts of quintic Jacobi sums and period polynomials for $\mathbf {F}_{ {q}}$

Abstract

In this paper, we construct explicit lifts of quintic Jacobi sums for finite fields via integer solutions of Dickson's system. Namely we give a procedure to compute quintic Jacobi sums for extended field $\mathbf{F}_{p^{s+t}}$ by using quintic Jacobi sums for $\mathbf{F}_{p^s}$ and for $\mathbf{F}_{p^t}$. We also have the multiplication formula from $\mathbf{F}_{p^s}$ to $\mathbf{F}_{p^{ns}}$ as a special case. By the quintuplication formula, we obtain the explicit factorization of the quintic period polynomials for finite fields.