Return to Revisions

Yes, it does hold in positive characteristic. You can show that the degree of $c_n(X)$ equals its Euler characteristic with respect to étale cohomology with coefficients in $\mathbb{Q}_{\ell}$, where $\ell$ is a prime different from the characteristic ( Top chern class in positive characteristic ), and this is has the required behavior under étale covers ( Behaviour of euler characteristics in characteristic p for finite etale covers ).