Is $\operatorname{Spec}( \mathbb Q[x_1\ldots,x_n])$ simply connected space? Can somebody point me good reference about the triviality of etale fundamental group for rational affine space over zero characteristic fields?
 A: $\mathrm{Spec}(\mathbb{Q}[x_1,...,x_n])=\mathbb{A}^n_\mathbb{Q}$ is not étale simply connected as $\widehat{\pi}_1(\mathbb{A}^n_\mathbb{Q})=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, but $\mathrm{Spec}(\overline{\mathbb{Q}}[x_1,...,x_n])=\mathbb{A}^n_{\overline{\mathbb{Q}}}$ is étale simply connected, as is the case for any algebraically closed field of characteristic 0. One way to see this is that for X a quasi-projective scheme over a field k we have a short exact sequence $0\rightarrow\widehat{\pi}_1(X_{\overline{k}})\rightarrow\widehat{\pi}_1(X)\rightarrow\mathrm{Gal}(\overline{k}/k)\rightarrow 0$, as any finite étale cover of $\mathrm{Spec}(k)$ induces a finite étale cover of $X$ by extension of scalars, which induces the trivial finite étale cover of $X_{\overline{k}}$ by extension of scalars to $\overline{k}$. In the case $X=\mathbb{A}^n_\mathbb{Q}$ we have $\widehat{\pi}_1(X_{\overline{\mathbb{Q}}})=0$ by a simple Riemann-Hurwitz argument so it follows that $\widehat{\pi}_1(\mathbb{A}^n_\mathbb{Q})=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ by exactness.
