Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Consider an unramified representation $\rho : Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p) \to \mathbb{F}_p^{\times}$ which sends the arithmetic Frobenius to an element $\mu$ \in \mathbb{F}_p$.

I know that the corresponding $(\varphi, \Gamma)$-module is of rank 1 over $\mathbb{F}_p((X))$ and that it admits a basis $e$ in which $\varphi(e) = 1 / \mu$ and the action of $\Gamma$ is trivial on $e$.

I realized that I am not able to prove it and it feels that it should be trivial. How can one obtain such a description ?