Let $k$ perfect field.
If we have a cubic non-singular projective curve $C(k)$ (over a field $k$), take two diferent points $P_1,P_2 \in C(k)$ and consider the line through the points, by Bezout theorem this line intersect to $C(\overline{k})$ in a unique third point $P_3$, now my question is:
Why the Galois Group $Gal(\overline{k}/k)$ allow us to say that $P_3 \in C(k)$?
The equations which define $P_3$ have coefficients in $k$, so $Gal(\bar k/k)$ preserves the locus of these equations i.e it fixes the coordinates of $P_3$ since it is the unique point of this locus.