We show that for any mod pm characters, Ο1β,β¦,Οkβ, the
Jacobi sum, x1β=1βpmββ―xkβ=1x1β+β―+xkβ=BββpmβΟ1β(x1β)β¦Οkβ(xkβ),
has a simple evaluation when m is sufficiently large (for mβ₯2 if
pβ€B). As part of the proof we give a simple evaluation of the mod pm
Gauss sums when mβ₯2