The Diagonal Quadratic Form And Jacobsthal Sum On GR(p²,p^2r)

Let R=G(p2,p2r)be a Galois ring.In this paper,we mainly study the number of solutions of diagonal quadratic forms over the Galois ring R in R*and extend the Jacobsthal sum to the Galois ring R,and we solve the number of solutions of some special equations in R*by using the related conclusions of the generalized Jacobsthal sum.In Chapter 1,we introduce some theories about the characteristic sums of Abelian groups,the conclusions of Jacobi sums over finite fields,the knowledge of Galois rings,and overview of the research and development of quadratic forms.In addition,we give main results of this paper.In Chapter 2,N(a1x12+a2x22+…+anxn2=b)denotes the number of solutions of a1x12+a2x22+…+anxn2=b in R*,where ai ∈ R\{0},b ∈ R.Using the method of solving the number of solutions of diagonal quadratic forms in field,We get the formulas of N(a1x12+a2x22+…+anxn2=b),ai ∈R*,b ∈ R and N(a1x12+a2x22+…+anxn2=b),ai∈ M\{0},b∈ R.In Chapter 3,we mainly obtain formulas about the Jacobsthal sum and its companion sum over the Galois ring R.By using the relevant conclusions,we give the formula for the number of solutions of the ternary cubic equation x2+y2+z2=2xyz in R*.