A User's Guide to Serre's Arithmetic/The Theorem on Arithmetic Progressions

Characters of Finite Abelian Groups

Modular Characters

Proposition 5 uses a couple clever tricks:

1. $\varepsilon :(2\mathbb {N} +1,\cdot )\to (\mathbb {Z} /2,+)$ is a monoid morphism
2. $x\equiv 1{\text{ }}({\text{mod }}4l_{1}\cdots l_{k})$ implies $x\equiv 1{\text{ }}({\text{mod }}4)$ and $x\equiv 1{\text{ }}({\text{mod }}l_{i})$