text: The case of univariate polynomials over a field is especially important for several reasons. Firstly, it is the most elementary case and therefore appears in most first courses in algebra. Secondly, it is very similar to the case of the integers, and this analogy is the source of the notion of Euclidean domain. A third reason is that the theory and the algorithms for the multivariate case and for coefficients in a unique factorization domain are strongly based on this particular case. Last but not least, polynomial GCD algorithms and derived algorithms allow one to get useful information on the roots of a polynomial, without computing them.
  Euclidean division
  Euclidean division of polynomials, which is used in Euclid's algorithm for computing  GCDs, is very similar to Euclidean division of integers. Its existence is based on the following theorem: Given two univariate polynomials a and b ≠ 0 defined over a field, there exist two polynomials q (the quotient) and r (the remainder)  which satisfy
  and
  where "" denotes the degree and the degree of the zero polynomial is defined as being negative. Moreover, q and r are uniquely defined by these relations.
  The difference from Euclidean division of the integers is that, for the integers, the degree is replaced by the absolute value, and that to have uniqueness one has to suppose that r is non-negative. The rings for which such a theorem exists are called Euclidean domains.