Polynomial ring/K/Greatest common divisor/Definition

Let ${\displaystyle {}P_{1},\ldots ,P_{n}\in K[X]}$ denote polynomials over a field ${\displaystyle {}K}$. We say that a polynomial ${\displaystyle {}G\in K[X]}$ is a greatest common divisor of the given polynomials, if ${\displaystyle {}G}$ is a common divisor of the ${\displaystyle {}P_{i}}$, and if ${\displaystyle {}G}$ has among all common divisors of the ${\displaystyle {}P_{i}}$ maximal degree.