text: In the previous section we have seen that the GCD of polynomials in  may be deduced from GCDs in  and in . A closer look on the proof shows that this allows us to prove the existence of GCDs in , if they exist in  and in . In particular, if GCDs exist in , and if  is reduced to one variable, this proves that GCDs exist in  (Euclid's algorithm proves the existence of GCDs in ).
  A polynomial in  variables may be considered as a univariate polynomial over the ring of polynomials in () variables. Thus a recursion on the number of variables shows that if GCDs exist and may be computed in , then they exist and may be computed in every multivariate polynomial ring over . In particular, if  is either the ring of the integers or a field, then GCDs exist in , and what precedes provides an algorithm to compute them.
  The proof that a polynomial ring over a unique factorization domain is also a unique factorization domain is similar, but it does not provide an algorithm, because there is no general algorithm to factor univariate polynomials over a field (there are examples of fields for which there does not exist any factorization algorithm for the univariate polynomials).