Valuation of a primitive p-power root of unity ($v({\zeta }_{p^a}-1)=v(p)/(p^{a-1}(p-1))$)

In a mixed-characteristic discretely valued field, a primitive $p^a$th root of unity reduces to one. For every $j$ prime to $p$, the ratio $({\zeta }^j-1)/(\zeta -1)$ is a unit, so all factors of ${\Phi }_{p^a}(1)=p$ have the same valuation. Taking valuations gives the formula. Because the left side is a positive integer for a normalized valuation, the existence of such a root forces $p^{a-1}(p-1)\mid v(p)$, bounding the possible $p$-power torsion.

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.