Topic summary
Coprime polynomials
Extracted from the Wikipedia article Polynomial greatest common divisor.
General definition
The greatest common divisor is unique only up to multiplication by an invertible constant. That is, if d is a GCD of p and q, then the polynomial c is another GCD if and only if there is an invertible element u of F such that In the case of the integers, this ambiguity may be removed by choosing the unique positive GCD, rather than its negative. For univariate polynomials over a field, one can instead choose the standard GCD to be the unique monic choice (having leading coefficient 1), but over more general coefficient rings there is no standard choice. Therefore, equalities like d = gcd(p, q) or gcd(p, q) = gcd(r, s) should be understood to mean "d is a GCD of p and q" and "p and q have the same set of GCDs as r and s". In particular, gcd(p, q) = 1 means that the invertible constants are the only common divisors: in this case, by analogy with the ring of integers, one says that p and q are coprime polynomials.
