Kummer cohomology divisibility criterion
= Kummer cohomology divisibility criterion
{c}
{title2=$H^{n+1}(L,\mu_p)=0$}
When the characteristic differs from $p$, the Kummer sequence gives $0\to H^n(L,K_s^{\times})/p\to H^{n+1}(L,\mu_p)\to H^{n+1}(L,K_s^{\times})[p]\to0$. For $n\ge1$, the positive-degree multiplicative groups are torsion. Thus the middle group is zero exactly when the first multiplicative group is $p$-divisible and the second has zero $p$-primary torsion.