Evaluating Prime Power Gauss and Jacobi Sums

We show that for any mod $p^m$ characters, ${\chi }_1,\dots ,{\chi }_k,$ the Jacobi sum, $$\sum _{x_1=1}^{p^m}\cdots \sum _{\begin{array}{c}{x}_k=1\\ x_1+\cdots +x_k=B\end{array}}^{p^m}{\chi }_1(x_1)\dots {\chi }_k(x_k),$$ has a simple evaluation when $m$ is sufficiently large (for $m\ge 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\ge 2$.