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

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 $$x \equiv 1 \text{ }(\text{mod } l_i)$$