In Kummer Theory, we always need the assumption that $a\not\in K^{\times n}$. For the local case, we can use the structure theorem of $K^{\times}$ to check this assumption. But in general, how can we check this condition?

For example, how to check $p\not\in K^{\times p}$, where $K=\mathbb{Q}(\mu_{p^\infty})$ is the rational number field with all the $p$th power roots of unity adjoined.