Bézout's Theorem
Theorem
Suppose that $X$ and $Y$ are 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
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Source of Name
This entry was named for Étienne Bézout.
He published it in his 1779 paper Théorie générale des équations algébriques.
