The Stacks project

Proof. Consider the fibre product $V = \mathop{\mathrm{Spec}}(k) \times _{U/G} U$. Here is a diagram

Then $V$ is a nonempty scheme étale over $\mathop{\mathrm{Spec}}(k)$ and hence is a disjoint union $V = \coprod _{i \in I} \mathop{\mathrm{Spec}}(k_ i)$ of spectra of fields $k_ i$ finite separable over $k$ (Morphisms, Lemma 29.36.7). We have

The action of $G$ on $U$ induces an action of $a : G \times V \to V$. The displayed equality means that $G \times V \to V \times _{\mathop{\mathrm{Spec}}(k)} V$, $(g, v) \mapsto (a(g, v), v)$ is an isomorphism. In particular we see that for every $i$ we have an isomorphism $H_ i \times \mathop{\mathrm{Spec}}(k_ i) \to \mathop{\mathrm{Spec}}(k_ i \otimes _ k k_ i)$ where $H_ i \subset G$ is the subgroup of elements fixing $i \in I$. Thus $H_ i$ is finite and is the Galois group of $k_ i/k$. We omit the converse construction. $\square$