Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer.
Consider $ g := x^n - 1 \in F[x]$
Is it true that $ g$ has distinct roots in $F$ if and only if $p$ does not divides $n$ ?
Here is my argument, please could you tell me whether it is correct:
Note roots of g are distinct if and only if $\mathrm{hcf}(g,g') = 1 $ .
Suppose $\beta \in F$ is a root of $g$ .
Now $ \beta^n = 1$, so $\beta \neq 0$ .
Then $g'(\beta) = n \beta^{-1} = 0 $ iff $p \mid n $
unless there is something wrong with how i am checking, i can't find any counterexamples for finite fields (where roots exist)
EDIT:
i think the statement is correct, at least i have found something which implies this in a book "Classical Galois Theory"
EDIT 2:
Noting $(a-b)^p = a^p - b^p $ in a field of characteristic p (what Dilip said)
then $$p \mid n \implies n = mp \implies x^n - 1 = (x^m-1)^p$$
and the polynomial on the RHS has only repeated roots. Can someone show me the other way?