有限域上n元二次方程的解数

The Solutions' Number of Quadratic Equations with n Variables over a Finite Field

设p为奇素数,Fp是含p个元素的有限域,n是大于1的自然数.通过方程的秩r,对Fp上n元二次方程进行了分类,若r=1,Fp上n元二次方程除(p-1)/2个外,其余都有解;若r>1,Fp上n元二次方程都有解,给出了各类方程的解数.

Let p be an odd prime, Fp be the finite field with p elements, and natural number n 〉 1 . By rank r of equations, quadratic equations with n variables over Fp is assorted. If r = 1 , there are solutions for quadratic equations with n variables over Fp besides （p - 1）/2 equations. If r 〉 1 ,there are solutions for quadratic equations with n variables over Fp . The numbers of solutions of quadratic equations with n variables over Fp have been given for every class equation.