Separability of a field Extension. Let $f=x^n-1$, and $L$ be a splitting field of $f$ over $K$ . 
Basically my question is to show that the extension $L|K$  is separable . 
Here is what i have been thinking , If char $K =0$ then its clear that its separable while $f' \not=0. 
$
 similarly in the case when Char$K =p$ where $p \not|  n$. 
But in the case when $p$ divides $n$ then we can write $n=p^km$ where $p\not|m$ 
then $x^n-1 =x^{p^km} -1 =(x^m-1)^{p^k}$ ie essentially the splitting field of $x^n-1$ is the splitting field of $x^m-1$, but splitting field of $x^m-1$ is separable with the 1st argument . 
Now here is my doubt $x^n-1$ has $n$ solution , but $x^m-1$ has $m$ solutions which is less than $n$  . 
i am confused whats going on here . Can someone help me or give me some example . 
Thanks for help .  
 A: (Let's actually record an answer!)
For any field $K$ and any positive integer $n$, the splitting field of $f = x^n-1$ is a separable field extension.  Because the polynomial $x^n-1$ is reducible for $n  > 1$ -- e.g. $x-1$ is a factor -- it may be cleanest not to reason with it directly.
Rather, the splitting field $L$ of $f$ is generated by adjoining to $F$ the $n$th roots of unity in an algebraic closure.  If we factor $n = \ell_1^{a_1} \cdots \ell_r^{a_r}$, we may separately adjoin the $\ell_i^{a_i}$ roots of unity: i.e., $L$ is the compositum of all these fields, so is separable iff each one is separable.
Case 1: $\ell$ is not equal to the characteristic of $K$.  Then the polynomial $x^{\ell_i^{a_i}} - 1$ has distinct roots in the algebraic closure, as can be checked by the usual Derivative Criterion (see e.g. $\S 6.1$ of my field theory notes), so its splitting field gives a separable field extension.
Case 2: $\ell = p$ is the characteristic of $K$.  In this case there are no nontrivial $p$-power roots of unity in $\overline{K}$, so the extension in question is just $K$ itself, which is of course separable (even though the polynomial $x^{p^n}-1 = (x-1)^{p^n}$ is not a separable polynomial according to the first definition discussed in the comments following the question).
