 Kummer cohomology divisibility criterion

When the characteristic differs from $p$, the Kummer sequence gives $0\to H^n(L,K_s^{\times })/p\to H^{n+1}(L,{\mu }_p)\to H^{n+1}(L,K_s^{\times })[p]\to 0$. For $n\ge 1$, 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.