What is $\sigma(\zeta)=\zeta^{\chi_l(\sigma)}$?

Question

What does the following mean ?

$$\sigma(\zeta)=\zeta^{\chi_l(\sigma)}$$

$\sigma\in {\rm Gal}(\bar{\mathbb Q}/\mathbb Q)$ and $\chi_l:{\rm Gal}(\bar{\mathbb Q}/\mathbb Q)\to\mathbb Z_l^{\times}$ and $\zeta$ is any $l^n$-th root of unity, how do you exponentiate that thing ?

I know Galois group, l-adic cyclotomic character and group actions in general.

Answer 1

If $\zeta$ is a $l^n$-th root of unity, then $a^{\chi_l(\sigma)}=a^{\chi_l(\sigma) \pmod{l^n}}$, where we used the projection map $\Bbb Z_l^\times \to (\Bbb Z/l^n\Bbb Z)^\times$.