Remarks

• The relevance of taking the coefficients in a field is that:
  • Nonzero coefficients are invertible, so that 2(X + 1) is a divisor of X² - 1 (in Q[X] say);
  • The product of two nonzero coefficients is itself nonzero, so that the product of two nonzero polynomials is nonzero as the product of their leading coefficients is nonzero. This is needed to guarantee uniqueness of the quotient (see next item).
• If the nonzero polynomial b divides a, then the quotient q is unique. For if qb = q1b, then (q - q1)b = 0. If q - q1 is nonzero, then the leading coefficient of the left-hand side is the nonzero product of the (nonzero) leading coefficients ofq - q1 and b. This contradicts the fact that (q - q1)b is the zero polynomial.
• If a and b are nonzero polynomials with coefficients in a field and if b divides a, then the degree of the quotient equals
  deg(a/b) = deg(a) - deg(b).