The Generalization Of Jacobi Sum In Matrix Form And Its Relation With Gauss Sum

In recent years,Kim[15],Li[5]et al.studied Gauss sums in the matrix form.Based on this,this paper analogically defines Jacobi sum in the matrix form Jn(χs,χt),and gives a conjecture that the connecting formula of Jacobi sum and Gauss sum is still established on the matrix ring.First,using the combination and number theory methods,this paper solves the calculation problem of 2-order Jacobi sum in the matrix form.Secondly,we use the matrix method to solve the computational problem of arbitrary non-trivial Jacobi sum.In order to make Jacobi sum more complete,we have also studied a class of bitrivial Jacobi sum.We first studied the 3-order bitrivial Jacobi sum.Using the methods of combination and multiple classification,the computational complexity of J3(χs,χt)can be reduced from q9 to q5.Based on some lower-order situations,we propose a recursive conjecture on the analytical formula of Jn(χs,χt),an n-order bitrivial Jacobi sum in the matrix form.Finally,we use the combination method to prove this conjecture.At this point,we completely solve the non-trivial Jacobi sum and bitrivial Jacobi sum in the matrix form.