For a semifield extension $T /S$, an action of a finite group $G$ on $T$ is Galois if $(1)$ the $G$-invariant subsemifield of $T$ is $S$ and $(2)$ subgroups of $G$ whose invariant semifields coincide are equal. We show that for a finite harmonic morphism between tropical curves $\varphi : \varGamma \to \varGamma^{ \prime}$ and an isometric action of a finite group $G$ on $\varGamma$, $\varphi$ is $G$-Galois if and only if the natural action of $G$ on the rational function semifield $\operatorname{ Rat}(\varGamma)$ of $\varGamma$ induced by the action of $G$ on $\varGamma$ is Galois for the semifield extension $\operatorname{ Rat}(\varGamma) / \varphi^{ \ast}(\operatorname{ Rat}(\varGamma^{ \prime}))$, where $\varphi^{ \ast}(\operatorname{ Rat}(\varGamma^{ \prime}))$ stands for the pull-back of $\operatorname{ Rat}(\varGamma^{ \prime})$ by $\varphi$.
Categories: math.AC math.AG