Bézout's identity

In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem:

Here the greatest common divisor of  and  is taken to be . The integers  and  are called Bézout coefficients for ; they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that  and  equality occurs only if one of  and  is a multiple of the other.

As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as  with Bézout coefficients −9 and 2.

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 all principal ideal domains.

Structure of solutions

If  and  are not both zero and one pair of Bézout coefficients  has been computed (for example, using the extended Euclidean algorithm), all pairs can be represented in the form

where  is an arbitrary integer,  is the greatest common divisor of  and , and the fractions simplify to integers.

If  and  are both nonzero, then exactly two of these pairs of Bézout coefficients satisfy 

and equality may occur only if one of  and  divides the other. 

This relies on a property of Euclidean division: given two non-zero integers  and , if  does not divide , there is exactly one pair  such that  and  and another one such that  and 

The two pairs of small Bézout's coefficients are obtained from the given one  by choosing for  in the above formula either of the two integers next to .

The extended Euclidean algorithm always produces one of these two minimal pairs.

Example

Let  and , then . Then the following Bézout's identities are had, with the Bézout coefficients written in red for the minimal pairs and in blue for the other ones.

If  is the original pair of Bézout coefficients, then  yields the minimal pairs via , respectively ; that is, , and .

Proof
Given any nonzero integers  and , let  The set  is nonempty since it contains either  or  (with  and ). Since  is a nonempty set of positive integers, it has a minimum element , by the well-ordering principle. To prove that  is the greatest common divisor of  and , it must be proven that  is a common divisor of  and , and that for any other common divisor , one has 

The Euclidean division of  by  may be written

The remainder  is in , because

Thus  is of the form , and hence  However,  and  is the smallest positive integer in : the remainder  can therefore not be in , making  necessarily 0. This implies that  is a divisor of . Similarly  is also a divisor of , and therefore  is a common divisor of  and .

Now, let  be any common divisor of  and ; that is, there exist  and  such that  and  One has thus

That is,  is a divisor of . Since  this implies

Generalizations

For three or more integers

Bézout's identity can be extended to more than two integers: if 

then there are integers  such that

has the following properties:
 d is the smallest positive integer of this form
 every number of this form is a multiple of d

For polynomials

Bézout's identity does not always hold for polynomials. For example, when working in the polynomial ring of integers: the greatest common divisor of  and  is x, but there does not exist any integer-coefficient polynomials p and q satisfying .

However, Bézout's identity works for univariate polynomials over a field exactly in the same ways as for integers. In particular the Bézout's coefficients and the greatest common divisor may be computed with the extended Euclidean algorithm.

As the common roots of two polynomials are the roots of their greatest common divisor, Bézout's identity and fundamental theorem of algebra imply the following result: 

The generalization of this result to any number of polynomials and indeterminates is Hilbert's Nullstellensatz.

For principal ideal domains
As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain (PID).
That is, if  is a PID, and  and  are elements of , and  is a greatest common divisor of  and ,
then there are elements  and  in  such that  The reason is that the ideal  is principal and equal to 

An integral domain in which Bézout's identity holds is called a Bézout domain.

History

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

See also
 , an analogue of Bézout's identity for homogeneous polynomials in three indeterminates

Notes

External links
 Online calculator for Bézout's identity.
 

Articles containing proofs
Diophantine equations
Lemmas in number theory