Polynomial ring/Field/Lemma of Bezout/Coprime/Fact

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}P_{1},\ldots ,P_{n}}$ denote coprime polynomials over ${\displaystyle {}K}$.

Then there exists a representation

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

with

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