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Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in ...
Claude Gaspar Bachet Sieur de Méziriac [1] (9 October 1581 – 26 February 1638) was a French mathematician and poet born in Bourg-en-Bresse, at that time belonging to Duchy of Savoy. [2]
In the case of plane curves, Bézout's theorem was essentially stated by Isaac Newton in his proof of Lemma 28 of volume 1 of his Principia in 1687, where he claims that two curves have a number of intersection points given by the product of their degrees. [3] However, Newton had stated the theorem as early as 1665. [4]
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that
A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal.
Proof: If d is this greatest common divisor, Bézout's identity asserts the existence of integers e and f such that ae + bf = d. If c is a multiple of d, then c = dh for some integer h, and (eh, fh) is a solution. On the other hand, for every pair of integers x and y, the greatest common divisor d of a and b divides ax + by.
This expression is called Bézout's identity. Numbers p and q like this can be computed with the extended Euclidean algorithm. gcd(a, 0) = | a |, for a ≠ 0, since any number is a divisor of 0, and the greatest divisor of a is | a |. [2] [5] This is usually used as the base case in the Euclidean algorithm.
In 1758 Bézout was elected an adjoint in mechanics of the French Academy of Sciences.Besides numerous minor works, he wrote a Théorie générale des équations algébriques, published at Paris in 1779, which in particular contained much new and valuable matter on the theory of elimination and symmetrical functions of the roots of an equation: he used determinants in a paper in the Histoire ...