On the Field-theoreticity of Homomorphisms between the Multiplicative Groups of Number Fields

We discuss the eld-theoreticity of homomorphisms between the multiplicative groups of number elds. We prove that, for instance, for a given isomorphism between the multiplicative groups of number elds, either the isomorphism or its multiplicative inverse arises from an isomorphism of elds if and only if the given isomorphism is SPU-preserving (i.e., roughly speaking, preserves the subgroups of principal units with respect to various nonarchimedean primes).