Remark 3

For every prime number l put μ_{l∞}(K) = U μ_{ln}(K). When l is the characteristic p of K, we have μ_{p∞}(K) = {1}. From I, p. 80, Th. 4, we deduce that μ_∞(K) is a direct sum of the subgroups μ_{l∞}(K), where l runs over the set of all prime numbers distinct from p. For a given prime number l only two cases are possible: either μ_{l∞}(K) is finite and then μ_{l∞}(K) is isomorphic to Z/l^nZ for a suitable n (Th. 1), or μ_{l∞}(K) is infinite and then μ_{l∞}(K) is isomorphic to Z[l^{-1}]/Z (cf. Remark 2).