Let $K$ be a splitting field of a polynomial over $\mathbb{Q}$. Suppose $K$ contains an $n$th root of some number $a$. Then how can we show that $K$ contains all the $n$th roots of unity?
I don't really see any relation between $\sqrt[n]{a}$ and the $n$th roots of unity. Knowing that the former is in $K$ doesn't seem to imply anything about the latter.