= Valuation of a primitive p-power root of unity
{title2=$v(\zeta_{p^a}-1)=v(p)/(p^{a-1}(p-1))$}
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.
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