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It's not true in general. Over a field of characteristic $p>0$, the map $f:\mathbf{P}^1\to\mathbf{P}^1$ f:\mathbf{A}^1\to\mathbf{A}^1$defined by$f(z)=z^p+z$is etale because its derivative is$1$. The degree of$f$is$p$, and the Euler characteristic of$\mathbf{P}^1$\mathbf{A}^1$ is $2$, 1$, but$2\neq 2p$1\neq 1\times p$.
It's not true in general. Over a field of characteristic $p>0$, the map $f:\mathbf{P}^1\to\mathbf{P}^1$ defined by $f(z)=z^p+z$ is etale because its derivative is $1$. The degree of $f$ is $p$, and the Euler characteristic of $\mathbf{P}^1$ is $2$, but $2\neq 2p$.