  We initiate the study of Nash blowups in prime characteristic. First, we show
that a normal variety is non-singular if and only if its Nash blowup is an
isomorphism, extending a theorem by A. Nobile. We also study higher Nash
blowups, as defined by T. Yasuda. Specifically, we give a characteristic-free
proof of a higher version of Nobile's Theorem for quotient varieties and
hypersurfaces. We also prove a weaker version for $F$-pure varieties.
