Let $k$ be a field of characteristic $p \geq 0$, $n$ an integer prime to $p$, and $x$ an element of $k \setminus \{0, 1\}$. I have read that the $n^{th}$ root of $1-x$ gives rise to a Galois module $E$ which is an extension
$$0 \to \mu_n \to E \to \mathbb{Z}/n\mathbb{Z} \to 0 $$
Could someone explain how this works? What is the geometric intuition behind?
I'm not sure if this is what you are looking for, but consider the Tate curve $E_q$ on your local field $k$. It sits on the exact sequence of $G_k$ modules.
$$0 \to \mu_n \to E_q[n] \to \mathbb{Z}/n\mathbb{Z} \to 0 $$
where $E_q[n]$ denotes the $n$-torsion points. And this gives us an exact sequence of Galois modules for the Tate module $T_n(E_q)$:
$$0 \to T_n(\mu_n) \to T_n(E_q) \to \mathbb{Z}_n \to 0 $$
Something similar works for Tate modules of general abelian varieties.
The geometric intuition boils down to Galois action and reduction $\mathrm{mod}$ $n$.
