Bézout's identity

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: Template:Math theorem

(The greatest common divisor of Template:Math and Template:Math is taken to be Template:Math.) The integers Template:Math and Template:Math are called Bézout coefficients for Template:Math; they are not unique. The extended Euclidean algorithm can be used to compute a minimal pair of Bézout coefficients, meaning they satisfy $|x|\le |b/d|$ and $|y|<|a/d|$; equality occurs only if one of Template:Math and Template:Math is a multiple of the other, and otherwise there exist exactly two minimal pairs.

As an example, the greatest common divisor of $a=15$ and $b=69$ is $d=3$, which can be written as the linear combination $ax+by=15(-9)+69(2)=3$ with Bézout coefficients $(x,y)=(-9,2)$, which are minimal since $9<69/3=23$ and $2<15/3=5$. The other minimal Bézout coefficients are $(x,y)=(-9+23,2-5)=(14,-3)$.

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 Template:Math and Template:Math are not both zero and one pair of Bézout coefficients Template:Math has been computed (for example, using the extended Euclidean algorithm), all pairs can be represented in the form $$(x-k\frac{b}{d},y+k\frac{a}{d}),$$ where Template:Math is an arbitrary integer, Template:Math is the greatest common divisor of Template:Math and Template:Math, and the fractions simplify to integers.

If Template:Mvar and Template:Mvar are both nonzero and none of them divides the other, then exactly two of the pairs of Bézout coefficients satisfy $$|x|<\left|\frac{b}{d}\right|\text{and}|y|<\left|\frac{a}{d}\right|.$$ If Template:Mvar and Template:Mvar are both positive, one has $x>0$ and $y<0$ for one of these pairs, and $x<0$ and $y>0$ for the other. If Template:Math is a divisor of Template:Mvar (including the case $b=0$), then one pair of Bézout coefficients is Template:Math.

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

The two pairs of minimal Bézout coefficients are obtained from the given one Template:Math by choosing for Template:Mvar in the above formula either of the two integers nearest to Template:Math.

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

Example

Let Template:Math and Template:Math, then Template:Math. 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.

$$\begin{array}{c}⋮\\ 12& \times (-10)& +42& \times 3& =6\\ 12& \times (-3)& +42& \times 1& =6\\ 12& \times 4& +42& \times (-1)& =6\\ 12& \times 11& +42& \times (-3)& =6\\ 12& \times 18& +42& \times (-5)& =6\\ ⋮\end{array}$$

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

Existence proof

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

The Euclidean division of Template:Mvar by Template:Mvar may be written as $$a=dq+r\text{with}0\le r<d.$$ The remainder Template:Mvar is in Template:Math, because $$\begin{array}{cc}r& =a-qd\\ & =a-q(as+bt)\\ & =a(1-qs)-bqt.\end{array}$$ Thus Template:Mvar is of the form Template:Math, and hence Template:Math. However, Template:Math, and Template:Mvar is the smallest positive integer in Template:Mvar: the remainder Template:Mvar can therefore not be in Template:Mvar, making Template:Mvar necessarily 0. This implies that Template:Mvar is a divisor of Template:Mvar. Similarly Template:Mvar is also a divisor of Template:Mvar, and therefore Template:Mvar is a common divisor of Template:Mvar and Template:Mvar.

Now, let Template:Mvar be any common divisor of Template:Mvar and Template:Mvar; that is, there exist Template:Mvar and Template:Mvar such that Template:Math and Template:Math. One has thus $$\begin{array}{cc}d& =as+bt\\ & =cus+cvt\\ & =c(us+vt).\end{array}$$ That is, Template:Mvar is a divisor of Template:Mvar. Since Template:Math, this implies Template:Math.

Corollaries

Writing any integer as a linear combination

An immediate corollary of Bézout's identity is that any integer Template:Mvar can be written as a linear combination of any two coprime integers. Indeed, if Template:Mvar and Template:Mvar are coprime, then Bézout's identity guarantees the existence of integers Template:Mvar and Template:Mvar such that Template:Math. Multiplying both sides by Template:Mvar gives $a(xn)+b(yn)=n$.

Generalizations

For three or more integers

Bézout's identity can be extended to more than two integers: if $$gcd(a_1,a_2,\dots ,a_n)=d$$ then there are integers $x_1,x_2,\dots ,x_n$ such that $$d=a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n$$ has the following properties:

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

For polynomials

Template:Main 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 Template:Math and Template:Math is Template:Math, but there are no integer-coefficient polynomials $p(x)$ and $q(x)$ satisfying $2xp(x)+x^{2}q(x)=x$.

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: Template:Block indent

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 Template:Math is a PID, and Template:Mvar and Template:Mvar are elements of Template:Math, and Template:Mvar is a greatest common divisor of Template:Mvar and Template:Mvar, then there are elements Template:Math and Template:Math in Template:Math such that Template:Math. The reason is that the ideal Template:Math is principal and equal to Template:Math.

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). 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

• Template:Annotated link, an analogue of Bézout's identity for homogeneous polynomials in three indeterminates
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External links

• Online calculator for Bézout's identity
• Template:Mathworld