I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 3 on page 660 reads as follows: (see attachment)

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Let in .

If is an element of V, then .

If , then also and we can write .

It follows that V is the set .

For any polynomial . we can write

... ... ... etc etc

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I cannot follow the line of reasoning:

"For any polynomial . we can write "

Can anyone clarify why this is true and why D&F are taking this step?

Peter