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 $\mathcal{k}\_{E}^{2}$.



Galois Action on the lines in the k^{2}_{E}
The action is fairly straightforward. Any line in $k\_E^2$ can be described by an equation of the form $ax+by=c$, where $a,b,c \in k\_E$, and $a$ and $b$ are not both zero. You can describe the action of $\operatorname{Gal}(E/F)$ on $k\_E$ by taking reduction modulo the maximal ideal. Alternatively, you can look at the action on the (prime-to-residue-characteristic) roots of unity in $E$ living over units in $k\_E$. You can then use the action on $k\_E$ to get the diagonal action on the coefficients $a,b,c$ in the equation $ax+by=c$.


If you want to restrict your view to lines through the origin, you get an action on pairs $(a,b)$, and the equivalence relation that sends proportional pairs to the same point in $\mathbb{P}^1$ is Galois-stable.


