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Polynomial ring over a field/One variable/Principal ideal domain/2/Fact/Proof

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Proof

Let denote an ideal in different from . We consider the non-empty set of natural numbers

This set has a minimum . This number arises from an element , , . We claim that .

The inclusion is clear. To prove the other inclusion , let be given. Due to fact, we have

Because of and the minimality of , the first case can not happen. Therefore, , and this means that is a multiple of .