= Roots of unity at the tame ramification boundary
For odd $p$, both possibilities occur at $v(p)=p-1$: $\mathbb Q_p(\zeta_p)$ contains $p$-torsion, while $\mathbb Q_p(\sqrt[p-1]{p})$ does not. Both extensions are totally ramified of degree $p-1$ by the <Eisenstein criterion>. In the second field, if $\zeta_p=1+u\pi$ with $\pi^{p-1}=p$, the cyclotomic equation reduced after division by $p$ would give $1+\bar u^{p-1}=0$ in $\mathbb F_p$, contradicting $\bar u^{p-1}=1$. For $p=2$, $v(2)=1$ always permits the root $-1$, so the torsion exponent is exactly one and the zero-exponent alternative is impossible.
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