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. $\epsilon :(2\mathbb{N}+1,\cdot )\to (\mathbb{Z}/2,+)$ is a monoid morphism
2. $x\equiv 1(\text{mod }4l_1\cdots l_k)$ implies $x\equiv 1(\text{mod }4)$ and $x\equiv 1(\text{mod }{l}_i)$