Borel fixed-point theorem fails for non-split linear algebraic groups

Short version
The analogue of the Borel fixed-point theorem for the case of an infinite non-algebraically closed field where the group is not a split algebraic group is not true.

Long version
For any infinite non-algebraically closed field $$K$$, there exists a solvable connected linear algebraic group $$G$$ over $$K$$ that is not a split algebraic group, such that there exists a projective variety $$V$$ over $$K$$ and an algebraic group action of $$G$$ on $$V$$ with no fixed point.

Example of the multiplicative group of a field extension
Since $$K$$ is not algebraically closed, it has finite-dimensional field extensions of degree strictly greater than one.

We consider the case that $$G$$ is the multiplicative group of a finite-dimensional field extension $$L$$ of $$K$$ of degree $$d > 1$$. Consider $$L$$ as a $$d$$-dimensional vector space over $$K$$. The action of $$G$$ on $$L$$ by multiplication defines an embedding of $$G$$ in $$GL(d,K)$$. This induces an action of $$G$$ on projective space $$V = \mathbb{P}^{d-1}(K)$$. We verify all the things:


 * $$G$$ is solvable: In fact, it is abelian
 * $$G$$ is connected: See algebraic torus is connected
 * $$G$$ is a linear algebraic group: This follows from its embedding in $$GL(d,K)$$
 * $$G$$ is not split: $$G = L^*$$ has a normal subgroup $$K^*$$ but the quotient group does not contain any copies of the additive or multiplicative group of $$K$$.
 * $$V$$ is a projective variety: By definition, projective spaces are projective varieties.
 * $$G$$ acts on $$V$$ algebraically (also called regularly): The action is a restriction of the natural action of $$GL(d,K)$$, which is algebraic.
 * The action has no fixed point: In fact, the action is transitive, and, if we think of $$G$$ as $$L^*$$, then $$V$$ is isomorphic to the quotient variety $$L^\ast/K^\ast$$.