For what $n$ are these polynomials separable?

Question

I already know that $P$ is separable iif $\gcd(P,P')=1$

I want to check for what $n$ these two polynomials are separable.

1. $P(X)=X^n-1\in \mathbb{Q}[X]$
2. $P(X)=X^n-1\in \mathbb{F}_4[X]$

For both polynomials we can compute $P'(X)= nX^{n-1}$. We see that the only factors of $P'$ are $X^k$ for $k=1,...,n-1$ but these are not factors of $P$ since it hasn't $0$ as root. So $\gcd(P,P')=1$ $\forall n$

For the second one I need to look at two cases.

2.1: $n\not \equiv 0\mod 4$, then it's the same reasoning as for (1).

2.2. $n\equiv 0\mod 4$ $\Rightarrow$ $P'(X) = nX^{n-1}=0$ $\Rightarrow$ $\gcd(P,P') = P$

Is that OK?