Let $p$ be a prime number. Let $\mathbb{F}$ be a finite extension of $\mathbb{F}_p$. Let $\omega$ be the mod $p$ cyclotomic character and let $V$ be a representation of $G_{p} = Gal(\bar{\mathbb{Q}}_p / \mathbb{Q}_p)$ over $\mathbb{F}$ which is a non-split extension of $\omega$ by $\omega$, namely we have a short exact sequence $1 \to \omega \to V \to \omega \to 1$.

We have a map $H^1 (G_p, V) \to H^1 (G_p, \omega)$ whose image is easily seen to be one dimensional over $\mathbb{F}$. Is it possible to describe it in terms of peu ramifiée or très ramifié extension ?

[EDIT] In order to be more precise, let me recall the definition of a peu ramifiée or très ramifiée extension in $H^1(G_p, \omega)$. (Those notions were introduced by Serre in Propriétés galoisiennes des points d'ordre fini des courbes elliptiques)

We have $H^1(G_p, \omega) \simeq \mathbb{Q}_p^{\times} / (\mathbb{Q}_p^{\times})^p \otimes_{\mathbb{F}_p} \mathbb{F}$ and the peu ramifiées extensions are the elements of the line $\mathbb{Z}_p^{\times} / (\mathbb{Z}_p^{\times})^p \otimes_{\mathbb{F}_p} \mathbb{F}$ and the très ramifiées ones are the complement of this set.

peu ramifiée and très ramifiéewhich have no equivalents in English. Applied to a (wildly) ramified degree-$p$ extension $L$ of a finite extension $K$ of $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$, they tell you whether the unique ramification break of $\mathrm{Gal}(L|K)$ is prime to $p$ or divisible by $p$. This is quite different from the distinction between tamely or wildly ramified extensions. $\endgroup$ – Chandan Singh Dalawat Aug 10 '13 at 1:27