Is Wiki wrong on Dirichlet Chararcters Modulo $10$?

Question

Wiki says:

Modulus 10

There are $\phi(10)=4$ characters modulo $10$.

Note that $χ$ is wholly determined by $\chi(3)$, since $3$ generates the group of units modulo $10$.

I can imagine that $\chi_2$ or $\chi_4$ generate $\chi$, by a certain kind of multiplication. But $3$? Is Wiki wrong? Maybe they started the table with $0$ once ago...

Answer 1

You're misunderstanding: the values of a Dirichlet character $\chi(n)$ for any $n$ are determined by $\chi(3)$, because $3$ is a generator of $(\mathbb{Z}/10\mathbb{Z})^\times$.

It is also true that the collection of Dirichlet characters modulo $m$ forms a group under (pointwise) multiplication, which in fact is isomorphic to $(\mathbb{Z}/m\mathbb{Z})^\times$. When $m=10$, we have $(\mathbb{Z}/10\mathbb{Z})^\times\cong\mathbb{Z}/4\mathbb{Z}$ and the characters listed in the table as $\chi_2$ and $\chi_4$ are generators.

Answer 2

I think you are mistaking $\chi(3)$ for $\chi_3$. It doesn't say $\chi_3$ generates anything. It says $\chi(3)$ determines $\chi(x)$ for all $x$.