Let $d \geq 2$ be an integer, $K$ a number field containing the $d$-th roots of unity $\mu_d(\mathbb{C})$ and $\mathfrak{p}$ a prime ideal of $K$ not dividing $d$. Let $\mathbb{F}_q$ be the residue field of $\mathfrak{p}$. I have seen in several places a multiplicative character $$ \chi: \mathbb{F}_q^\times \to \mu_d(\mathbb{C}) $$ defined by $$ \chi(x)=Teich_{\mathfrak{p}}(x^{(q-1)/d}) $$ where $Teich_{\mathfrak{p}}: \mu_d(\mathbb{F}_q) \to \mu_d(\mathbb{C})$ is the inverse of reduction mod $\mathfrak{p}$.
For this to make sense, we need that reduction mod $\mathfrak{p}$ of roots of unity gives an isomorphism (then it follows that $d$ divides $q-1$). But why is this true?
Also, is there any relation between the Teichmüller character here and the one for $p$-adic numbers?
Thanks for your help!