关于p^k(p>3)元域上的三次方程

On the Cubic Equations over a Field of p^k(p>3)Elements

F是pk(p >3 )元域 .本文首先证明 ,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 ) ;而后得到 ,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 ;给出了必要充分条件 。

In this paper,let F be a field of p k(p>3)elements.First,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,he 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 diffrent roots and it has no roots in the field F.He gives the necessary and sufficient conditions and completely answers this problem.