I know that to prove $\pi_1(\mathbb{P}_k^n) = 0$, one can just use the curve case and proceed by induction. For example, if $n=2$ and there is any connected etale covering, one can choose any line in $\mathbb{P}_k^2$. The inverse image of the line is the support of an ample divisor hence connected by Hartshorne III.7.9. Then one can conclude the covering is of degree 1 and finish the proof. But I don’t know how to do this in the affine case and I don’t want to use comparison theorem to the topological fundamental group. Can someone tell me how to do this? Thank you very much!
I’m not sure if the following argument works. One can prove that if $k$ is not required to be algebraic closed, then all connected finite etale coverings of $\mathbb{A}_k^n$ are $\mathbb{A}_{k’}^n$ where $k’/k$ is a finite separable extension. Now we can do induction as following. When $k$ is algebraic closed, If $f:X\rightarrow Spec(k[x,y])$is a connected finite etale covering, then so is the base change$f’:X’\rightarrow Spec(k(x)[y])$. By induction hypothesis, $X’=Spec(L[y])$ where $L$ is a finite separable extension of $k(x)$. Hence the generic fiber of $y=0$ is $Spec(L)$ which is connected. Then one can use induction hypothesis to conclude $X=\mathbb{A}_k^n$. When k is not algebraic closed, I guess it can be deduced from the precious case.