Consider the case where $\displaystyle F =

Z_p $ and the polynomial $\displaystyle X^{p^k} - X $

splits over K containing F . Show that the set

$\displaystyle GF_{p^k} =

\{c \in K : c^{p^k}

= c \} $ has exactly $\displaystyle p^k $ elements.

I tried showing $\displaystyle X^{p^k} - X $

has $\displaystyle p^k $ distinct roots but did not succeed.