Let $p$ be a prime number, $\mathbb Q_p$ be the completion of $\mathbb Q$ w.r.t $\,p$, denote $\xi_n$ primitive $n$-th root of unity in a fixed algebraic closure of $\mathbb Q_p$ with $p\nmid n$, we have:
If $\mathbb Q_p(\xi_n) = \mathbb Q_p$, then $n|(p-1)$.
Why is the above property true?
I know a property that if $L$ unramified over $\mathbb Q_p$, then $L\simeq \mathbb Q_p(\xi_{p^m-1})$ where $m= [L:\mathbb Q_p]$, using this we get $\mathbb Q_p =\mathbb Q_p(\xi_n)\simeq \mathbb Q_p (\xi_{p-1}).$ Then I'm not sure how to proceed.