We show that for any mod 2m characters, χ1​,χ2​, the complete
exponential sum, x=1∑2m​χ1​(x)χ2​(Axk+B), has a simple
explicit evaluation
We show that for any mod pm characters, χ1​,…,χk​, the
Jacobi sum, x1​=1∑pm​⋯xk​=1x1​+⋯+xk​=B​∑pm​χ1​(x1​)…χk​(xk​),
has a simple evaluation when m is sufficiently large (for m≥2 if
p∤B). As part of the proof we give a simple evaluation of the mod pm
Gauss sums when m≥2