Editing Bézout's identity (section)

== Generalizations ==

=== For three or more integers===

Bézout's identity can be extended to more than two integers: if 
<math display="block">\gcd(a_1, a_2, \ldots, a_n) = d</math>
then there are integers <math>x_1, x_2, \ldots, x_n</math> such that
<math display="block">d = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n</math>
has the following properties:
* {{math|''d''}} is the smallest positive integer of this form
* every number of this form is a multiple of {{math|''d''}}

=== For polynomials ===

{{main|Polynomial greatest common divisor#Bézout's identity and extended GCD algorithm}}
Bézout's identity does not always hold for polynomials. For example, when working in the [[polynomial ring]] of integers: the greatest common divisor of {{math|2''x''}} and {{math|''x''<sup>2</sup>}} is ''x'', but there does not exist any integer-coefficient polynomials {{math|''p''}} and {{math|''q''}} satisfying {{math|1=2''xp'' + ''x''<sup>2</sup>''q'' = ''x''}}.

However, Bézout's identity works for [[univariate polynomial]]s over a [[field (mathematics)|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 [[root of a polynomial|root]]s of two polynomials are the roots of their greatest common divisor, Bézout's identity and [[fundamental theorem of algebra]] imply the following result: 
{{block indent|em=1.5|text=For univariate polynomials {{mvar|f}} and {{mvar|g}} with coefficients in a field, there exist polynomials {{math|''a''}} and {{math|''b''}} such that {{math|1=''af'' + ''bg'' = 1}} if and only if {{mvar|f}} and {{mvar|g}} have no common root in any [[algebraically closed field]] (commonly the field of [[complex number]]s).}}

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 (algebra)|ring]] of integers, but also in any other [[principal ideal domain]] (PID).
That is, if {{math|''R''}} is a PID, and {{mvar|a}} and {{mvar|b}} are elements of {{math|''R''}}, and {{mvar|d}} is a greatest common divisor of {{mvar|a}} and {{mvar|b}},
then there are elements {{math|''x''}} and {{math|''y''}} in {{math|''R''}} such that {{math|1=''ax'' + ''by'' = ''d''}}. The reason is that the [[ideal (ring theory)|ideal]] {{math|''Ra'' + ''Rb''}} is principal and equal to {{math|''Rd''}}.

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