The Sum Set Of Weighted Exceptional Units And The Solution Of The Diagonal Quadratic Form On The Galois Ring

This article is divided into three parts.The main research is about the sum-sets of weighted exceptional units in a finite commutative ring and the number of solutions of diagonal quadratic forms over Galois rings GR(p2,p2m).The paper is organized as follows:In Chapter 1,we introduce the definition of finite communicative rings with identity and Galois rings,the backgrounds of the representation of weighted excep-tional units and quadratic diagonal forms.Let R be a finite commutative ring with identity.In Chapter 2,we obtain a formula for the number of ways to represent each element of R to the sum of some weighted exceptional units.Let R=GR(p2,p2m)be a Galois ring and N(a1x12+…+anxn2＝b)denote the number of solutions of quadratic diagonal forms a1x12+…+anxn2=b,where a1,…,an∈R*,x1,…,xn,b∈R.Our main result in chapter 3 is to get formulae of N(ax2 =b),N(a1x12+ a2x22= b)and N(a1x12+ a2x22+a3x32= b).