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The Riemann-Hurwitz formula implies that the projective line $\mathbb{P}^1_K$ over any algebraically closed field $K$ is simply connected (i.e., $\pi_1^{et}(\mathbb{P}^1_K) = 1$; equivalently, if $\...

In the proof of Corollary 10.12 of Exposé I of SGA 1 something like the following is asserted as a ‘known lemma’: Let $k$ be an infinite field and $B$ a finite $k$-algebra. If $B$ is not a product ...