text: The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of the Euclidean algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need for efficiency of computer algebra systems.
  General definition
  Let  and  be polynomials with coefficients in an integral domain , typically a field or the integers.
  A greatest common divisor of  and  is a polynomial  that divides  and , and such that every common divisor of  and  also divides . Every pair of polynomials (not both zero) has a GCD if and only  if  is a unique factorization domain.
  If  is a field and  and  are not both zero, a polynomial  is a greatest common divisor if and only if it divides both  and , and it has the greatest degree among the polynomials having this property. If , the GCD is 0. However, some authors consider that it is not defined in this case.
  The greatest common divisor of  and  is usually denoted "".
  The greatest common divisor is not unique: if  is a GCD of  and , then the polynomial  is another GCD if and only if there is an invertible element  of  such that