Unique primitive ancestor of a Dirichlet character

Write $q=\prod p^{e_p}$. For each $p$, choose the least $c_p\le e_p$ for which the local unit character factors through reduction modulo $p^{c_p}$; the finite set of possible exponents has such a least element.

CRT combines the resulting local characters into a character ${\chi }^{\ast }$ modulo $d=\prod p^{c_p}$, and [F1] shows that its induction has the same values as $\chi$, including zero precisely on the nonunits modulo $q$.

If a primitive character modulo $d^{\mathrm{\prime }}$ induces $\chi$, each of its local exponents must be at least $c_p$; minimality gives $d\mid d^{\mathrm{\prime }}$. If $d^{\mathrm{\prime }}>d$, its local character factors through a proper divisor, contradicting primitivity. Hence $d^{\mathrm{\prime }}=d$ and CRT gives ${\chi }^{\ast }={\chi }^{\ast \mathrm{\prime }}$.