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I am reading Fontaine and Yi's note "Theory of $p$ adic Galois Representations,

http://www.math.u-psud.fr/~fontaine/galoisrep.pdf

I got confused in section 1.2.1.

Say $K$ is a finite field of characteristic $p$ with $q$ elements, let $K^s$ be a fixed algebraic closure of $K$ and $G=\text{Gal}(K^s/K) \cong \widehat{\mathbb{Z}}$ be the Galois group. Let $K_n$ be the unique extension of $K$ of degree $n$ inside $K^s$ for $n \geq 1$ and $\tau_K(x)=x^{q^{-1}}$ be the geometric Frobenius automorphism.

Then an $l$-adic representation $\rho:G \rightarrow \text{Aut}_{\mathbb{Q}_l}(V)$ is given by \begin{equation} \rho(\tau_K)=u,\,u\in\text{Aut}_{\mathbb{Q}_l}(V) \end{equation} Then every element of $G$ is of the form $\tau_K^n,\,n\in \widehat{\mathbb{Z}}$, then how to define $\rho(\tau_K^n)$?(I do not know what the limit means in the note)!

I also do not know how to prove Prop 1.10 and Prop 1.11 in the note! Anyone who can provide more details?

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