Remark 1

When K is an algebraically closed field of characteristic 0, the group μ_∞(K) is thus isomorphic (not canonically) to Q/Z. When K is the field C of complex numbers, we can write down an explicit such isomorphism: in effect the mapping x ↦ e^{2πix} is a homomorphism of Q into C with kernel Z and image μ_∞(C); thus by passage to quotients it defines an isomorphism of Q/Z onto μ_∞(C).*