Can Bezout's theorem be generalized to non algebraically closed fields?

Question

The Bezout's theorem says that the intersection of two curves in $\mathbb{P}^2_k$, (counting multiplicity, $k$ is algebraically closed) is equal to the product of their degrees. Can the theorem be generalized to non algebraically closed field? Where the intersection points are replaced by the intersection of schemes?

Answer 1

Let $C, D$ be two curves in $\mathbb P^2_k$, without commun irreducible component. For any point $x\in C\cap D$, define the multiplicity $\mathrm{mult}_x(C.D)$ (over $k$) of $C\cap D$ as the dimension over $k$ of $O_{\mathbb P^2_k, x}/(f, g)$, where $f, g\in O_{\mathbb P^2_k, x}$ are the respective local equations of $C, D$ at $x$. Then $$ \deg C \deg D=\sum_{x\in C\cap D} \mathrm{mult}_x(C.D).$$ It would be better to define mult$_x$ as the length over $O_{\mathbb P^2,x}$ of the quotient ring, but then in the above formula, one has to multiply mult$_x$ by the degree $[k(x): k]$ of the residue field at $x$.