| For mod p Dirichlet characters chi1, chi2 the classical Jacobi sums [special characters omitted] have a long history in number theory. In particular, it is well known that if chi1, chi2 and chi1chi2 are non-trivial characters, then J(chi1, chi2, p) can be written in terms of Gauss sums and |J(chi1, chi2, p)|=p1/2, though in general no evaluation is known without the absolute value. In this thesis we consider some mod pm generalization of the Jacobi sums where we can obtain an explicit evaluation (without the absolute value) for m sufficiently large. For example, if chi, chi1, ..., chi s are mod pm Dirichlet characters the sums [special characters omitted] where p | A1... As,B,k1 ... ks, and [special characters omitted], where p |2A1... AsB(1--w 1-... --ws), have simple evaluations when m ≥ 2. Exponential or character sums with an explicit evaluation are rare. Interestingly the sums we consider here can, like the classical Jacobi sums, be written in terms of Gauss sums. |
