On the number of solutions of some Kummer equations over finite fields

Let l   be a prime number and let k=Fqk=Fq be a finite field of characteristic p≠lp≠l with q=pfq=pf elements. Let n≥0n≥0. We determine the number N   of solutions (x,y)(x,y) in k of the Kummer equationyl=x(xln−1),yl=x(xln−1), in terms of the trace of a certain Jacobi sum.