# Kollár/Smith/Corti 一章, 問題1.9

Let $k'/k$ be a purely inseparable extension of degree $p^a$. Let $C$ be a smooth curve in $\mathbb{A}^n_{k'}$ and let $P$ be a $k'$-point of $C$. Prove that the divisor $p^aP$ can be defined by polynomials in $k[x_1,\ldots,x_n]$.

Proof. Let $f = \sum a_Ix^I \in k'[C]$ vanish at $P$ with multiplicity one. Then, $f^{p^a} = \sum a_I^{p^a}x^I \in k[C]$ vanishes at $P$ with multiplicity $p^a$.