P^K(P>3)元域上的三次方程根的状况

The Roots of the Cubic Equations over a Field of P~ K(P>3) Elements

F是一个PK(P >3)元域 .本文证明 :研究F上的三次方程可以转化为研究方程x3+ax +b =0 (a≠ 0 ,b≠ 0 ) .然后得到x3+ax +b =0 (a≠ 0 ,b≠ 0 )在域F中有且仅有一根 ,或一个单根与一个二重根 ,或三个互异的根 ,或没有根 .最后 。

Let F be a field of P~ K(P>3) elements.In this paper,the author proves that studying any cubic equation can change into studying equation x 3+ax+b=0(a≠0,b≠0) over the Field F.Afterwards,it is obtainted that the equation x 3+ax+b=0(a≠0,b≠0) has one and only one root or one simple root and one twofold root or three different roots,and it has no roots in the field F.Finally,the author completely gives the roots of the cubic equations over a finite field.