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