The gcd of two positive integers is the greatest among all divisors,
both in the absolute sense and with respect to the (partial) ordering
given by division.
Here follows a similar characterization for polynomials, where the degree
is the measure for size.
Proposition
Let a, b, c be three
polynomials in R[X]. If a and b are not both zero
and c is a common divisor of a and
b of maximal degree, then c is a
greatest common divisor of a and
b.