| Gauss sums and Jacobi sums are two important classes of character sums.They have been widely used in both theoretical and applied mathematics.Classical Gauss sums and Jacobi sums are defined over residue class rings of integers or finite fields.Recently,people started to study Gauss sums and Jacobi sums defined on some other finite rings with identity due to theoretical and practical needs.By far,Gauss sums and Jacobi sums defined on a finite commutative ring with identity has been well understood.For non-commutative rings,only matrix rings over finite fields were studied incompletely.In this thesis,we consider the Gauss sums and Jacobi sums defined on matrix rings over residue class rings.We will define Gauss sums and Jacobi sums over these rings and try to evaluate them.We will see that there are a lot of imprimitive cases in these cases,compared to the case defined over matrix rings over finite fields.We show that these sums in imprimitive cases are either zero or reduced to the sums in primitive cases,and primitive Gauss sums and Jacobi sums have similar properties with classical ones.In particular,we generalize the work of Dong [23] and prove the conjectures he proposed. |
