I am currently trying to understand a step in a proof concerning an étale morphism. Unfortunately I lack a deep understanding of this concept. I hope someone can elucidate that.
Let $\phi: \mathbb{A}_{\mathbb{F}_p}^k \to G$ be a separable morphism. Denote by $J\in \mathbb{F}[t_1, \ldots,t_k]$ the Jacobian to this morphism as follow. Let $X_1, \ldots, X_k$ be the basis of $(\text{Lie } G)^*$ and then extend the $X_i$ to global right-invariant one-forms $\bar{X}_i$ on $G$ and finally define $J$ by $$\phi^*(\bar X_1 \wedge \ldots \bar X_k) =J(dt_1 \wedge \ldots \wedge dt_k)$$ Assume that $J(0) = 1$ and denote by $W$ the principal open subset of $\mathbb{A}_{\mathbb{F}_p}^k$ given by $J\neq 0$. Then $\phi|_W$ is étale.
Can someone explain how this follows?
Thanks in advance.