 Solution

Every finite extension of degree prime to $p$ is an algebraic extension. Thus the universal vanishing established for all algebraic extensions in (ii) holds in particular for the extensions appearing in (iii). Equivalently, the Kummer reduction gives $H^{n+1}(L,{\mu }_p)=0$ for each of these finite extensions. Hence (ii) implies (iii). No transfer or separability assumption is required for this implication.