I'm currently working through Elliptic Curves: Number Theory and Cryptography, L. C. Washington.
The bottom of page 82 says "Since the characteristic of $K$ does not divide $n$, the equation $x^n = 1$ has no multiple roots, hence has $n$ roots in $\overline{K}$."
Why is this true? I'm aware that if a polynomial $f(x)$ has a root which is also a root to $f'(x)$, then that root is a multiple root, but I feel like I'm missing a step to deduce the sentence.
Let $f(x)=x^n-1$. Then $f'(x)=nx^{n-1}$. If the characteristic of $K$ does not divide $n$, then $f'\ne0$ and so $\gcd(f,f')=1$.
Conversely, if the characteristic of $K$ is $p$ and $n=mp$, then $x^n-1=x^{mp}-1=(x^m-1)^p$. Therefore, every root of $x^m-1$ is a multiple root of $x^n-1$.