Abstract: Starting with an action $G\times X\to X$ we analyze the maximal $G$-rational subalgebra $\mathscr{O}_K$ of $k(X)$ and use it to obtain the action $G_K\times U_K\to U_K$ where $K = k(X)^G$, and $U_K$ is a certain quasi-affine variety over $K$ with $\mathscr{O}(U_K) = \mathscr{O}_K$. This gives us a generic "homogeneous" picture of the original action.

We also analyze the maximal $G$-rational subalgebra of $k[X]_\mathfrak{p}$, where $\mathfrak{p}$ is a height-one $G$-prime of $k[X]$. We use these results to assess the behavior of the canonical map $\pi : U\to U/G$ for a sufficiently small $G$-invariant, open subset $U$ of $X$.

Finally we use ${\textit{observable}}$ $G$-actions over $k$ to construct the functor $K\mapsto H^1(K,G/H)$, from finitely generated fields over $k$ to ${\textit{Sets}}$. From there we define the ${\textit{essential dimension}}$ of a homogeneous space $G/H$, whenever $H\subset G$ is a pair of connected, reductive groups.