# Thread: Polynomial Roots

1. ## Polynomial Roots

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.

2. Hi

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

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

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

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