text: The proof that the algorithm satisfies its output specification relies on the fact that, for every  we have
  the latter equality implying
  The assertion on the degrees follows from the fact that, at every iteration, the degrees of  and  increase at most as the degree of  decreases.
  An interesting feature of this algorithm is that, when the coefficients of Bezout's identity are needed, one gets for free the quotient of the input polynomials by their GCD.
  Arithmetic of algebraic extensions
  An important application of the extended GCD algorithm is that it allows one to compute division in algebraic field extensions.
  Let  an algebraic extension of a field , generated by an element whose minimal polynomial  has degree . The elements of  are usually represented by univariate polynomials over  of degree less than .
  The addition in  is simply the addition of polynomials:
  The multiplication in  is the multiplication of polynomials followed by the division by :
  The inverse of a non zero element  of  is the coefficient  in Bézout's identity , which may be computed by extended GCD algorithm. (the GCD is 1 because the minimal polynomial  is irreducible). The degrees inequality in the specification of extended GCD algorithm shows that a further division by  is not needed to get deg() < deg().