# Bézout's Theorem

## Theorem

Let $X$ and $Y$ be two plane projective curves defined over a field $F$ that do not have a common component.

(This condition is true if both $X$ and $Y$ are defined by different irreducible polynomials. In particular, it holds for a pair of "generic" curves.)

Then the total number of intersection points of $X$ and $Y$ with coordinates in an algebraically closed field $E$ which contains $F$, counted with their multiplicities, is equal to the product of the degrees of $X$ and $Y$.

## Proof

## Source of Name

This entry was named for Étienne Bézout.

## Historical Note

**Bézout's Theorem** was originally published in $1799$ by Étienne Bézout in his *Théorie Générale des Équations Algébriques*.