Another proof of a theorem on rational cross sections

Abstract

The extant proofs of the existence of a rational cross section for a transformation space for a connected solvable linear algebraic group either use a certain amount of algebraic curve theory or restrict themselves to the case of a principal space, where the question is one of galois cohomology, the result being equivalent to the statement that H X(G, k) = 0 for G a A>solvable linear algebraic group. The present proof of the general result may be considered more elementary in that it depends only on the standard facts on fields of rationality of algebraic sets. The result in question says that if G is a k-solvable linear