# Bézout's Theorem

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## 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

## Also known as

Some sources omit the accent off the name: **Bezout's theorem**, which may be a mistake.

## Source of Name

This entry was named for Étienne Bézout.

## Historical Note

**Bézout's Theorem** was originally published in $1779$ by Étienne Bézout in his *Théorie Générale des Équations Algébriques*.

## Sources

- 1779: Étienne Bézout:
*Théorie Générale des Équations Algébriques* - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Bezout's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Bézout's Theorem**