I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets, Example 2 on page 660 reads as follows: (see attachment)

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(2) Over any field k, the ideal of functions vanishing at is a maximal ideal since it is the kernel of the surjective ring homomorphism from to the field k given by evaluation at .

It follows that

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I can see that gives zeros for each polynomial in - indeed, to take a specific example involving we have for, let us say, a particular polynomial where g is as follows:

so in this case, clearly ... ... ... and, of course, other polynomials in similarly.

BUT ... ... I cannot understand D&Fs reference to maximal ideals. Why is it necessary to reason about maximal ideals.

Since I am obviously missing something, can someone please help by explaining what is going on in this example.

Would appreciate some help.

Peter