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If you have non-Arch. local field F and E a finite extension, I am just wondering if anybody have an 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 $\mathcal{k}_{E}^{2}$.

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

If you have non-Arch. local field F and E a finite extension, I am just wondering if anybody have an 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.
• Tree of $SL(2,F) \simeq \mathbb{P}_F^1$

If you have non-Arch. local field F and E a finite extension, I am just wondering if anybody have an 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 $\mathcal{k}_{E}^{2}$.