= Smooth point on an irreducible hypersurface in positive characteristic
Let $k$ be algebraically closed of characteristic $p>0$ and let $f\in k[x_1,\ldots,x_n]$ be irreducible. Some partial derivative of $f$ is nonzero: otherwise every exponent is divisible by $p$, and perfection of $k$ would make $f$ a $p$th power. If the gradient vanished at every point of $Z(f)$, the <Strong Hilbert Nullstellensatz> would put each partial derivative in $(f)$, impossible by degree. Hence $Z(f)$ has a smooth point and minimum tangent dimension $n-1$.
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