Polynomial ring/Field/Lemma of Bezout/Fact

Lemma of Bezout (polynomial ring)

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

Then there exists a representation

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

with

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

Proof, Write another proof