In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is a theorem which relates two arbitrary integers with their greatest common divisor. The theorem's statement is as follows:

{{math_theorem

| name = Bézout's identity

| math_statement = Let and be integers with greatest common divisor . Then there exist integers and such that . Moreover, the integers of the form are exactly the multiples of .

}}

(The greatest common divisor of and is taken to be .) The integers and are called Bézout coefficients for ; they are not unique. The extended Euclidean algorithm can be used to compute a minimal pair of Bézout coefficients, meaning they satisfy and ; equality occurs only if one of and is a multiple of the other, and otherwise there exist exactly two minimal pairs.

As an example, the greatest common divisor of and is , which can be written as the linear combination with Bézout coefficients , which are minimal since and . The other minimal Bézout coefficients are .

Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, can be formally deduced 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 and none of them divides the other, then exactly two of the pairs of Bézout coefficients satisfy

If and are both positive, one has and for one of these pairs, and and for the other. If is a divisor of (including the case ), then one pair of Bézout coefficients is .

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 minimal Bézout coefficients are obtained from the given one by choosing for in the above formula either of the two integers nearest 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] held, with the Bézout coefficients written in red for the minimal pairs and in blue for the other ones.

<math display="block">\begin{align}

\vdots \\

12 &\times ({\color{blue}{-10}}) & + \;\; 42 &\times \color{blue}{3} &= 6 \\

12 &\times ({\color{red}{-3}}) & + \;\;42 &\times \color{red}{1} &= 6 \\

12 &\times \color{red}{4} & + \;\;42 &\times({\color{red}{-1}}) &= 6 \\

12 &\times \color{blue}{11} & + \;\;42 &\times ({\color{blue}{-3}}) &= 6 \\

12 &\times \color{blue}{18} & + \;\;42 &\times ({\color{blue}{-5}}) &= 6 \\

\vdots

\end{align}</math>

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

Existence 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 as

The remainder is in , because

<math display="block">\begin{align}

r & = a - qd \\

& = a - q(as+bt)\\

& = a(1-qs) - bqt.

\end{align}</math>

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

<math display="block">\begin{align}

d&=as + bt\\

& =cus+cvt\\

&=c(us+vt).

\end{align} </math>

That is, is a divisor of . Since , this implies .

Corollaries

Writing any integer as a linear combination

An immediate corollary of Bézout's identity is that any integer can be written as a linear combination of any two coprime integers. Indeed, if and are coprime, then Bézout's identity guarantees the existence of integers and such that . Multiplying both sides by gives .

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:

  • is the smallest positive integer of this form
  • every number of this form is a multiple of

For polynomials

Bézout's identity does not always hold for polynomials with coefficients in a ring. For example, when working in the polynomial ring with integer coefficients: the greatest common divisor of and is , but there are no integer-coefficient polynomials and satisfying .

However, Bézout's identity works for univariate polynomials with coefficients in a field, exactly as for the original case of 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 the 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 and attribution

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

</ref> On these pages, Bachet proves (without equations) "Proposition XVIII. Deux nombres premiers entre eux estant donnez, treuver le moindre multiple de chascun d’iceux, surpassant de l'unité un multiple de l'autre." (Given two numbers [which are] relatively prime, find the lowest multiple of each of them [such that] one multiple exceeds the other by unity (1).) This problem (namely, ) is a special case of Bézout's equation and was used by Bachet to solve the problems appearing on pages 199 ff.See also:

Andrew Granville traced the association of Bézout's name with the identity to Bourbaki, arguing that it is a misattribution since the identity is implicit in Euclid's Elements.

See also

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

References

  • Online calculator for Bézout's identity