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

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Proof

We consider the set of all linear combinations

This is an ideal of , as can be checked directly. Due to fact, this ideal is a principal ideal, hence,

with a certain polynomial . This is a common divisor of the . Because of , we have

that is is a factor of every . A similar reasoning shows

for all , and, therefore, also

Hence,

By the condition, has the maximal degree among all common divisors. Therefore, is a constant. Thus, we have

and, in particular, . Therefore, is a linear combination of the .