  We initiate the study of the resolution of singularities properties of Nash
blowups over fields of prime characteristic. We prove that the iteration of
normalized Nash blowups desingularizes normal toric surfaces. We also introduce
a prime characteristic version of the logarithmic Jacobian ideal of a toric
variety and prove that its blowup coincides with the Nash blowup of the
variety. As a consequence, the Nash blowup of a, not necessarily normal, toric
variety of arbitrary dimension in prime characteristic can be described
combinatorially.
