simple extension

[Al Don Gate]

Hi,

 

Let [math]L/K[/math] be a simple extension : [math]L/K[/math] is a a field extension and it exits [math]x\in L[/math] such that L=K(x).

 

I consider the endomorphism of L [math]m_x[/math] define by [math]m_x(y)=xy[/math].

 

My problem :

 

I find ( and i can explain if somebody want...) that the characteristic polynomial of m_x is 0 !!! It involves that the degree of the extension is 0??? How is it possible?