This should hopefully be a very simple question:
"Suppose $\mbox{char}K = 0$ or $p$, where $p\not| \ m $. The $m$th cyclotmic extension of $K$ is just the splitting field $L$ over $K$ of $X^m - 1$"
Is the condition that $p \not | \ m$ there to guarantee $X^m - 1$ is separable over $K$? (the derivative is $mX^{m-1}$, which certainly shares a factor of degree $\geq 1$ with $X^m - 1$ if $m$ is a multiple of $p$)
EDIT: A follow up question:
$L/K$ is Galois. Why does an element $\sigma$ in $\mbox{Gal}(L/K)$ send primitive roots to primitive roots?