Let $k$ be a field of characteristic zero, and let ${\overline k}$ be an algebraic closure of $k$ . For a geometrically integral variety $X$ over $k$ , we write ${\overline k}(X)$ for the function field of ${\overline X}=X\times_k{\overline k}$ . If $X$ has a smooth $k$ -point, the natural embedding of multiplicative groups ${\overline k}^*\hookrightarrow {\overline k}(X)^* $ admits a Galois-equivariant retraction.
¶ In the first part of this article, equivalent conditions to the existence of such a retraction are given over local and then over global fields. Those conditions are expressed in terms of the Brauer group of $X$ .
¶ In the second part of the article, we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For $k$ local or global, and for such a variety $X$ , in many situations but not all, the existence of a Galois-equivariant retraction to ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ 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.