Proposition 17

Let $n\geqslant 1$ be an integer. Let us denote by $\boldsymbol{\mu}_n$ the group of $n$-th roots of unity in $\mathbf{C}$. The groups $\mathbf{Z}/n\mathbf{Z}$ and $\boldsymbol{\mu}_n$ are in duality relative to the mapping induced by passing to the quotient from the mapping $\mathbf{Z}\times \boldsymbol{\mu}_n\rightarrow \mathbf{U}$ defined by $(m, z)\mapsto z^m$.

The group $\mathbf{Z}\widehat{/n}\mathbf{Z}$ coincides with the set of homomorphisms $\chi$ of $\mathbf{Z}/n\mathbf{Z}$ into $\mathbf{U}$. These are of the form $m\mapsto \chi (1)^m$, where $\chi (1)$ is an arbitrary element of $\mathbf{U}$ such that $\chi (1)^n= 1$, whence the result.