# Borel fixed-point theorem

A more general version of the theorem holds over a field k that is not necessarily algebraically closed. A solvable algebraic group G is split over k or k-split if G admits a composition series whose composition factors are isomorphic (over k) to the additive group ${\displaystyle \mathbb {G} _{a}}$ or the multiplicative group ${\displaystyle \mathbb {G} _{m}}$. If G is a connected, k-split solvable algebraic group acting regularly on a complete variety V having a k-rational point, then there is a G fixed-point of V.[1]