What is the étale fundamental group of projective space over an algebraically closed field?


In char = 0 it is trivial (Lefschetz principle), as well as in dimension 1 (Riemann-Hurwitz).



étale fundamental group of projective space
It is a birational invariant (for smooth proper connected schemes over a field, ultimately due to Zariski-Nagata purity of the branch locus), and its formation is compatible with products (for proper connected schemes over an algebraically closed field), so we can replace projective $n$-space with the $n$-fold product of copies of the projective line to conclude. Likewise, due to limit arguments and invariance of the etale site with respect to finite radiciel surjections (such as a finite purely inseparable extension of a ground field), it suffices to take the ground field to be separably closed rather than algebraically closed. This is all in SGA1. 


