Let $p$ be a prime number $\geq 3$. Let $V$ be a representation of $Gal(\bar{\mathbb{Q}}\_p/ \mathbb{Q}\_p)$ with coefficients in $\mathbb{F}\_p$. Assume $V$ is a non-split extension of two characters $\delta\_1$ and $\delta\_2$.


Let $D$ be the $(\varphi, \Gamma)$-module associated to $V$. It is a non-split extension of two $(\varphi, \Gamma)$ of rank $1$. Denote by $\Phi$ the matrix of $\varphi$ and by $G$ the matrix of a generator $\gamma$ of $\mathbb{Z}\_p^\*$. In a suitable basis, we have $\Phi = \begin{pmatrix} f\_1 & f \\ 0 & f\_2 \end{pmatrix}$ and $G = \begin{pmatrix} g\_1 & g \\ 0 & g\_2 \end{pmatrix}$ where $f\_1, f\_2, g\_1$ and $g\_2$ are elements of $\mathbb{F}\_p^{\times}$ (see [One dimensional (phi,Gamma)-modules in char p](https://mathoverflow.net/questions/15902/one-dimensional-phi-gamma-modules-in-char-p)) and where $f,g \in \mathbb{F}\_p ((X))$.


Can we find a basis where $f$ and $g$ are in $\mathbb{F}\_p[[X]]$ ? If not, what is the "nicest" form possible for $f$ and $g$ ? (i.e. can we kill some of the denominators and which one ?)



$(\varphi, \Gamma)$-module of dimension 2 modulo $p$
The answer is "no" in general. If $\delta\_2=1$ and $\delta\_1$ is the mod $p$ cyclotomic character, then there are both peu and très ramifiées extensions. Let $res(g)$ denote the coefficient of $1/X$ in $g$. Theorem: in your notation, the extension is peu ramifiée iff $res(g)=0$. See for instance proposition 3.7.5 of Mathieu Vienney's PhD <http://perso.ens-lyon.fr/laurent.berger/thesevienney.pdf>


As to the "nicest" possible form: you can construct elements of $H^1(G\_{Q\_p},V)$ from elements of $D(V)^{\psi=1}$, see lemma I.5.5 of Cherbonnier and Colmez' JAMS paper. If you make this explicit, you'll have strong restrictions on $f$ and $g$.


