Polynomial ring/Field/1/Common divisor/Section
Let denote polynomials over a field . We say that a polynomial is a common divisor of the given polynomials if divides
every .![{\displaystyle {}P_{1},\ldots ,P_{n}\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e8674da56efd0ecc032b0bb15b57a84730bd32)
![{\displaystyle {}T\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4849e575dad5bbdf92559d0de92981d221c74c)
Let denote polynomials over a field . We say that a polynomial is a greatest common divisor of the given polynomials, if is a common divisor of the , and if has among all common divisors of the maximal
degree.![{\displaystyle {}G\in K[X]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4ecbefea2e6d4f6016a1b0fb1c29bc8a5f6e7d)
A greatest common divisor is not uniquely determined, since with also for some constant is also a greatest common divisor. However, if we restrict to normed polynomials, then the greatest common divisor is unique.
Polynomials over a field are called coprime, if they have, with the exception of constants , no
common divisor.