Polynomial ring/Field/Lemma of Bezout/Fact

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}P_{1},\ldots ,P_{n}}$ denote polynomials over ${\displaystyle {}K}$. Let ${\displaystyle {}G}$ be a greatest common divisor of the ${\displaystyle {}P_{i}}$.

Then there exists a representation

${\displaystyle {}G=Q_{1}P_{1}+\cdots +Q_{n}P_{n}\,}$

with

${\displaystyle {}Q_{1},\ldots ,Q_{n}\in K[X]}$.