Solution
= 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 \b[(ii) implies (iii)]. No transfer or separability assumption is required for this implication.