# Math Help - Polynomial Roots

1. ## Polynomial Roots

Consider the case where $F =
Z_p$
and the polynomial $X^{p^k} - X$
splits over K containing F . Show that the set
$GF_{p^k} =
\{c \in K : c^{p^k}
= c \}$
has exactly $p^k$ elements.
I tried showing $X^{p^k} - X$
has $p^k$ distinct roots but did not succeed.

2. Hi

Since $D(X^{p^k} - X)=-1,\ X^{p^k} - X$ has only distinct roots

3. Here is another way to prove this that does not use derivatives.

Let $F$ be a field with charachteristic $p$ then the polynomial $x^n - 1$ has $n$ roots in a splitting field if $p\not |n$.

We can write, if $a$ is a zero then we can write $x^n - 1 = (x-a)(x^{n-1}+x^{n-2}a+...+xa^{n-2}+a^{n-1})$. But the second polynomial, $x^{n-1}+...+a^{n-1}$, when evaluated at $a$ is $a^{n-1}+a^{n-1}+...+a^{n-1} = n\cdot a^{n-1} \not = 0$ since $p\not | n$.