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Coefficients of the characteristic polynomial of the map on algebraic de Rham cohomology

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Coefficients of the characteristic polynomial of the map on de Rham cohomology

Let $k$ be an algebraically closed field. Let $V$ be a smooth projective variety over $k$. For a map $\phi:V\to V$, do the coefficients of the characteristic polynomial of $\phi^*:H^ i_{dR}(V/k)\to H^ i_{dR}(V/k)$ lie in the prime subfield of $k$?