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The greatest common divisor g of two nonzero numbers a and b is also their smallest positive integral linear combination, that is, the smallest positive number of the form ua + vb where u and v are integers.
For two univariate polynomials p and q over a field, there exist polynomials a and b, such that (,) = + and (,) divides every such linear combination of p and q (Bézout's identity). The greatest common divisor of three or more polynomials may be defined similarly as for two polynomials.
A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal.
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 3 = 15 × (−9) + 69 × 2, 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.
In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
A third difference is that, in the polynomial case, the greatest common divisor is defined only up to the multiplication by a non zero constant. There are several ways to define unambiguously a greatest common divisor. In mathematics, it is common to require that the greatest common divisor be a monic polynomial.
An integral domain is a UFD if and only if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals. An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two.
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