On Poincaré sums for number fields (Q816831)

Let \(k\) be a number field and \(K/k\) a finite normal extension with Galois group G. Fix a one dimensional cocycle \(c\in Z(G,O_K^{\times})\) and set \(M_c = \{x\in O_K^{\times}\mid c(s)s(x)=x\) for\(s\in G\}\), \(P_c = \{\sum_{s\in G}c(s)s(x)\mid x\in O_K^{\times}\}\). Then \(P_c\subseteq M_c\). The author shows that the index of \(P_c\) in \(M_c\) is the product of the corresponding local indices.