Bézout's identity

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Bézout's identity

In number theory, Bézout's identity for two integers a, b is an expression $ax+by=d \,$

where x and y are integers (called Bézout coefficients for (a,b)), such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers (a,b). In addition d > 0 is their greatest common divisor and the smallest positive integer that can be written, in this form, for any integers x,y. This value of d is therefore uniquely determined by a and b, but the Bézout coefficients are not unique. A pair of Bézout coefficients (in fact the ones that are minimal in absolute value) can be computed by the extended Euclidean algorithm. The identity, coefficients and lemma are named after the French mathematician Étienne Bézout.

In abstract algebra, certain pairs of elements of an integral domain may also admit such an identity, but this need not be the case for all pairs of nonzero elements. Bézout's lemma does however remain valid in any principal ideal domain.

History

Étienne Bézout (1730–1783) proved this identity for polynomials. However, this statement for integers can be found already in the work of French mathematician Claude Gaspard Bachet de Méziriac (1581–1638).

Algorithm

The Bézout numbers x and y as above can be determined with the extended Euclidean algorithm. However, they are not unique. If one solution is given by (x, y), then there are infinitely many solutions. These are given by $\left\{ \left(x+\frac{kb}{\gcd(a,b)},\ y-\frac{ka}{\gcd(a,b)}\right) \mid k \in \mathbb{Z} \right\}.$

Example

The greatest common divisor of 12 and 42 is 6. Bézout's identity states that there must exist an integer solution for x and y in the following equation: $12x + 42y = 6. \,$

One of its solutions is x = −3 and y = 1: indeed, we have (−3)·12 + 1·42 = 6. Another solution is x = 4 and y = −1.

Generalizations

Bézout's identity can be extended to linear combinations of more than two numbers. For any numbers $a_1, \ldots, a_n$ with greatest common divisor d there exist integers $x_1, \ldots, x_n$ such that $a_1 x_1 + \cdots + a_n x_n = d.$

The greatest common divisor of $a_1, \ldots, a_n$ is in fact the smallest positive integer that can be written as a linear combination of $a_1, \ldots, a_n$.

Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain (PID). That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b, then there are elements x and y in R such that ax + by = d. The reason: the ideal Ra+Rb is principal and indeed is equal to Rd. An integral domain in which Bézout's identity holds is called a Bézout domain.

Proof

A proof of Bézout's lemma can be given using of the fact that the integers form a Euclidean domain (for the absolute value), as affirmed by the defining property of Euclidean division of integers, namely that the remainder of a division by a nonzero integer b has a remainder strictly less than |b|. For given nonzero integers a and b there is a nonzero integer d = as + bt of minimal absolute among all those of the form ax + by with x and y integers; one can assume d > 0 by changing the signs of both s and t if necessary. Now the remainder of dividing either a or b by d is also of the form ax + by since it is obtained by subtracting a multiple of d = as + bt from a or b, and on the other hand it has to be strictly smaller in absolute value than d. This leaves 0 as only possibility for such a remainder, so d divides a and b exactly. If c is another common divisor of a and b, then c also divides as + bt = d, which means that d is the greatest common divisor of a and b; this completes the proof.

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