Suppose I have $\chi$, a Dirichlet character modulo $q$. Let $q = p_1^{e_1} \cdots p_r^{e_r}$ be the prime factorization of $q$. Then do there exist characters $\chi_j$ modulo $p_j^{e_j}$ for each $1 \leq j \leq r$ such that $\chi = \chi_1 \cdots \chi_r$? And furthermore is this decomposition unique?
I thought the answer was yes for both questions but my neighbor insists that there is a problem when $2|q$ and I wanted to verify this with someone... I would greatly appreciate clarification on this. Thank you very much.