Suppose $p$ is a prime, $\chi$ and $\lambda$ are characters on $\mathbb{F}_p$. How can I show that $\sum_{t\in\mathbb{F}_p}\chi(1-t^m)=\sum_{\lambda}J(\chi,\lambda)$ where $\lambda$ varies over all characters such that $\lambda^m=id$?
($J$ is the Jacobi sum, defined as $J(\chi,\lambda)=\sum_{a+b=1}\chi(a)\lambda(b)$ )
The exercise is taken from A Classical Introduction to Modern Number Theory by Ireland and Rosen - page 105, ex. 8.