On rational multiplicative group actions

We establish a one-to-one correspondence between rational multiplicative group actions on an algebraic variety $X$ and derivations $\mathrm{\partial }\text{\hspace{0.17em}⁣}:K_X\to K_X$ of the field of fractions $K_X$ of $X$ satisfying that there exists a generating set $\{a_i{\}}_{i\in I}$ of $K_X$ as a field such that $\mathrm{\partial }(a_i)={\lambda }_i{a}_i$ with ${\lambda }_i\in \mathbb{Z}$ for all $i\in I$. We call such derivations rational semisimple. Furthermore, we also prove the existence of a rational slice for every rational semisimple derivation, i.e., an element $s\in K_X$ such that $\mathrm{\partial }(s)=s$. By analogy with the case of additive group actions case, we prove that $K_X\simeq K_X^{{\mathbb{G}}_m}(s)$ and that under this isomorphism the derivation $\mathrm{\partial }$ is given by $\mathrm{\partial }=s\frac{d}{ds}$. Here, $K_X^{{\mathbb{G}}_m}$ is the field of invariant of the ${\mathbb{G}}_m$-action.