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

We consider the set of all linear combinations

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

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

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

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

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

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

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

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

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

Hence,

${\displaystyle {}E=GF\,.}$

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

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

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