Ramification bound for a primitive pth root of unity
= Ramification bound for a primitive pth root of unity
{title2=$e(L/\mathbb Q_p)\geq p-1$}
If a finite extension of the <P-adic numbers> contains a primitive $p$th <root of unity> $\zeta$, the identity $p=\prod_{j=1}^{p-1}(1-\zeta^j)$ gives $e=(p-1)v_L(1-\zeta)\geq p-1$. The ratios of these factors reduce to the nonzero integers $j$ in the <residue field>. Thus an extension with $e<p-1$ has only roots of unity of order prime to $p$.