Explicit lifts of quintic Jacobi sums and period polynomials for Fq

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.