If you have non-Arch. local field F and E its finite extension, I am just wondering if anybody has any idea about the action of $\operatorname{Gal}(E/F)= \operatorname{Aut}_F(E)$ on the lines in $k^2_E$, Where $k_E$ is a residue field of the extension?

Note:

- $\{ \text{Adjacent points} \} \simeq \mathbb{P}^1_k$
- $\mathbb{P}_{1}(k):=$ set of all dim 1 subspaces of $k\mathcal{P}^{1}$ a 2-dim k-vector space.
- $\mathbb{E}\mathcal{P^{1}}=$ lines in $\mathcal{k}_{E}^{2}$.

dividesq-1. Then the normal basis theorem implies that for every character x:Gal(l|k)-->k^*, the x-eigenspace l(x) is a k-line. – Chandan Singh Dalawat Oct 7 '10 at 3:17