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

Let $K$ be a field and let $P_1,\dots ,P_n$ denote coprime polynomials over $K$.

Then there exists a representation

$Q_1{P}_1+\cdots +Q_n{P}_n=1$

with

$Q_1,\dots ,Q_n\in K[X]$.

Proof, Write another proof