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

Proof

We consider the set of all linear combinations

${}I={\left\{Q_{1}P_{1}+\cdots +Q_{n}P_{n}\mid Q_{i}\in K[X]\right\}}\,.$

This is an ideal of ${}K[X]$, as can be checked directly. Due to fact, this ideal is a principal ideal, hence,

${}I=(E)\,$

with a certain polynomial ${}E$. This ${}E$ is a common divisor of the ${}P_{i}$. Because of ${}P_{i}\in I=(E)$, we have

${}P_{i}=H_{i}E\,,$

that is ${}E$ is a factor of every ${}P_{i}$. A similar reasoning shows

${}P_{i}\in (G)\,$

for all ${}i$, and, therefore, also

${}(E)=I\subseteq (G)\,.$

Hence,

${}E=GF\,.$

By the condition, ${}G$ has the maximal degree among all common divisors. Therefore, ${}F\neq 0$ is a constant. Thus, we have

${}(G)=(E)=I\,,$

and, in particular, ${}G\in I$. Therefore, ${}G$ is a linear combination of the ${}P_{i}$.

To fact