 Inflection point of a plane cubic

A flex of a smooth plane cubic is a point whose tangent line has intersection multiplicity three with the cubic. Over an algebraically closed field, choose a flex $O$ as the elliptic curve identity. A line cuts out a divisor linearly equivalent to $3O$, so $P$ is a flex exactly when $3P=O$: the tangent divisor is then $3P$. Thus the flexes are precisely the geometric 3-torsion points.

If the characteristic is not three, prime-to-characteristic geometric torsion gives nine distinct flexes. In characteristic three, the scheme-theoretic fibre $E[3]$ still has length nine, but need not have nine distinct points: an ordinary elliptic curve has three geometric 3-torsion points, and a supersingular elliptic curve has only the identity. A flex chosen over the algebraic closure need not be rational over the original field.