  Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure
of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar
k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a
smooth $k$-point, the natural embedding of multiplicative groups ${\bar
k}^*\hookrightarrow {\bar k}(X)^* $ admits a Galois-equivariant retraction.
  In the first part of the paper, over local and then over global fields,
equivalent conditions to the existence of such a retraction are given. They are
expressed in terms of the Brauer group of $X$.
  In the second part of the paper, we restrict attention to varieties which are
homogeneous spaces of connected but otherwise arbitrary algebraic groups, with
connected geometric stabilizers. For $k$ local or global, for such a variety
$X$, in many situations but not all, the existence of a Galois-equivariant
retraction to ${\bar k}^*\hookrightarrow {\bar k}(X)^* $ ensures the existence
of a $k$-rational point on $X$. For homogeneous spaces of linear algebraic
groups, the technique also handles the case where $k$ is the function field of
a complex surface.
