Galois group action via character

Question

Let $G_K$ be a absolute Galois group of field $K$. Let $f$ be character of $G_K$, then, what does

'$G_K$ act via $f$'

mean ? (I searched the definition, but I couldn't find exactly fits this context).

context:I encountered this word when searching the example of one dimensional p adic representation.

Answer 1

For $K=\Bbb{Q}$ the natural $p$-adic character of $G_K$ is, for $\sigma \in G_K$, the unique $\chi(\sigma)\in \Bbb{Z}_p^\times$ such that $\sigma(\zeta_{p^n})=\zeta_{p^n}^{\chi(\sigma)\bmod p^n}$ for all $n$.

The corresponding $G_K$-action on $\Bbb{Q}_p$ is $\sigma\cdot a=\chi(\sigma)a$ for all $a\in \Bbb{Q}_p$.