Bézout

Bézout refers to a concept in mathematics named after the French mathematician Étienne Bézout. It is most commonly associated with Bézout's identity, a fundamental theorem in number theory. Bézout's identity states that if d is the greatest common divisor of two integers a and b, then there exist integers x and y such that ax + by = d. These integers x and y are known as Bézout coefficients. The identity is crucial for understanding divisibility properties and solving linear Diophantine equations.

The extended Euclidean algorithm is a practical method for computing the greatest common divisor of two integers and simultaneously finding the corresponding Bézout coefficients. This algorithm is widely used in various fields, including cryptography, particularly in the RSA algorithm, where it is essential for calculating modular multiplicative inverses. The existence of Bézout coefficients for any pair of integers guarantees that certain algebraic structures, like the integers modulo n, behave well under operations related to gcd. The concept can also be generalized to polynomial rings, where a similar identity holds for polynomials.