Smooth plane cubic in characteristic three
= Smooth plane cubic in characteristic three
{title2=$Y^2Z-X^3+XZ^2=0$}
Over the algebraic closure of $\mathbb F_3$, the displayed cubic is smooth. Its three partial derivatives are $Z^2$, $2YZ$, and $Y^2+2XZ$; their simultaneous vanishing would give $Z=Y=0$, and the equation would then give $X=0$, impossible in projective space. Its affine equation $y^2=x^3-x$ is irreducible because the odd-degree polynomial on the right is not a square in the rational function field.