p^k元域上的方程∑a^ix^(n-1-i)=0与∑(-a)~ix^(n-1-i)=0

The equations ∑a^ix^(n-1-i)=0 and ∑(-a)~ix^(n-1-i)=0 over a field with p^k elements

F是一个pk元域,n是一个正整数.xn-1+axn-2+…+an-2x+an-1=0(a≠0)与xn-1-axn-2+…+(-a)n-2x+(-a)n-1=0(a≠0)是F上的方程.本文完整地给出这些方程在F中的根的状况:(n,pk-1)-1个单根,(n,pk-1)组互不相同的重根,没有根.同时,给出根的求法及例子.

In this paper,let F be a field with p^k elements and n be a positive integer.The x^(n-1)+ax^(n-2)+…+a^(n-2)x+a^(n-1)=0(a≠0) and x^(n-1)-ax^(n-2)+…+(-a)^(n-2)x+(-a)^(n-1)=0(a≠0) are equations over the F.The author completely gives the roots of these equations in the F,that is,they have (n,p^k-1)-1 single roots or (n,p^k-1) distinct groups of repeated roots,and they have no root.At the same time,he gives the method and examples of finding roots.