I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ over $F_p$ which are in correspondence with 1-dimensional (phi,Gamma)-modules over $F_p((T))$.

There are finitely many such Galois representations. Moreover, their associated (phi,Gamma)-modules are very simple -- the action of phi and Gamma can be described (on some basis element) as scaling by an element of $F_p$ (as opposed to $F_p((T))$).

My question: can one see this directly on the (phi,Gamma)-module side? That is, given a 1-dimensional (phi,Gamma)-module $D$ over $F_p((T))$, find a $D'$ isomorphic $D$ such that $D'$ has a basis in which the matrices for phi and elements of Gamma are in $F_p$.

I would like to better understand the simplest case of the correspondence between Galois representations and (phi,Gamma)-modules. Namely, consider 1-dimensional Galois representations of $G_{Q_p}$ over $F_p$ which are in correspondence with 1-dimensional etale (phi,Gamma)-modules over $F_p((T))$.

There are finitely many such Galois representations. Moreover, their associated (phi,Gamma)-modules are very simple -- the action of phi and Gamma can be described (on some basis element) as scaling by an element of $F_p$ (as opposed to $F_p((T))$).

My question: can one see this directly on the (phi,Gamma)-module side? That is, given a 1-dimensional etale (phi,Gamma)-module $D$ over $F_p((T))$, find a $D'$ isomorphic $D$ such that $D'$ has a basis in which the matrices for phi and elements of Gamma are in $F_p^\times$.