# Bézout's Theorem

Jump to navigation
Jump to search

## 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.)

This page or section has statements made on it that ought to be extracted and proved in a Theorem page.In particular: Link to a proof of the above, and (preferably) move this statement to the proof itself.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by creating any appropriate Theorem pages that may be needed.To discuss this page in more detail, feel free to use the talk page. |

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

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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):**Bezout's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Bézout's Theorem**