What points of affine space can be mapped to zero by an étale morphism? Let $K$ be a field and $n$ a positive integer.
For what points $x\in\mathbb{A}^n_K$ can I find an étale morphism $f_x:\mathbb{A}^n_K\to \mathbb{A}^n_K$ mapping $x$ to zero and how does such a morphism look like on the level of $K$-algebras $g_x:K[X_1,\ldots,X_n]\to K[X_1,\ldots,X_n]$?
Edit: By ''points'' $x\in\mathbb{A}^n_K$ I mean ''elements of $\operatorname{Spec} K[X_1,\ldots,X_n]$'', all prime ideals.
 A: A necessary condition for $f_x:\mathbb{A}^n_K\to \mathbb{A}^n_K$ to be étale at $x$ is that the extension $K\to \kappa(x)$ be finite separable .
This implies that $x$ is a closed point of $\mathbb{A}^n_K$, but not necessarily a rational one.  
For example the morphism $f:\mathbb{A}^1_\mathbb Q\to \mathbb{A}^1_\mathbb Q$ induced by $\mathbb Q[T]\to \mathbb Q[X]: T\mapsto X^2-2$ maps the non-rational point $x\in \mathbb{A}^1_\mathbb Q$ corresponding to the prime ideal $(X^2-2)\in \mathbb Q[X]$ to the origin   $0\in \mathbb{A}^1_\mathbb Q$ [corresponding to $(T)\in \operatorname{Spec}(\mathbb Q[T])$] of the codomain of $f$.
This morphism $f$ is étale at $x$, but not globally étale since it is not étale at the origin $(X)\in \operatorname{Spec} (\mathbb Q[X])=\mathbb A^1_\mathbb Q$ of the  domain of the morphism $f$.
I think the search for globally étale  morphisms as in your question might be quite difficult, since even automorphisms of affine space are intractable, as witnessed by the notorious Jacobian conjecture.    
