The Idelic Approach to Class Groups and Unit Groups

Notation: Despite some misgivings, we introduce the following notation. For a global field K, we let ΣK denote the set of all places of K, i.e., equivalence classes of nontrivial norms on K. (Recall that at this point we have associated a canonical normalized Artin absolute value to each place in K.) Further we write Σ K for the subset of non-Archimeden places of K, and let Σ K denote the subset of Archimedean places of K, which is nonempty iff K is a number field.