I am reading Dummit and Foote Ch 15, Commutative Rings and Algebraic Geometry. In Section 15.1 Noetherian Rings and Affine Algebraic Sets,(adsbygoogle = window.adsbygoogle || []).push({}); Example 2 on page 660reads as follows: (see attachment)

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(2) Over any field k, the ideal of functions vanishing at [itex] (a_1, a_2, ... ... ... a_n) \in \mathbb{A}^n [/itex] is a maximal ideal since it is the kernel of the surjective ring homomorphism from [itex] k[x_1, x_2, ... ... x_n] [/itex] to the field k given by evaluation at [itex] (a_1, a_2, ... ... ... a_n) [/itex].

It follows that [itex] \mathcal{I}((a_1, a_2, ... ... ... a_n)) = (x - a_1, x - a_2, ... ... ... , x - a_n) [/itex]

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I can see that [itex] (x - a_1, x - a_2, ... ... ... , x - a_n) [/itex] gives zeros for each polynomial in [itex] k[ \mathbb{A}^n ] [/itex] - indeed, to take a specific example involving [itex] \mathbb{R} [x,y] [/itex] we have for, let us say, a particular polynomial [itex] g \in \mathbb{R} [x,y] [/itex] where g is as follows:

[itex] g(x,y) = 6(x - a_1)^3 + 11(x - a_1)^2(y - a_2) + 12(y - a_2)^2 [/itex]

so in this case, clearly [itex] g(a_1, a_2) = 0 [/itex] ... ... ... and, of course, other polynomials in [itex] \mathbb{R} [x,y] [/itex] 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.

Another issue I have is why do D&F write [itex] \mathcal{I}((a_1, a_2, ... ... ... a_n)) [/itex] with 'double' parentheses and not just [itex] \mathcal{I}(a_1, a_2, ... ... ... a_n) [/itex]?

Would appreciate some help.

Peter

Note - see attachment for definition of [itex] \mathcal{I}(A) [/itex]

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# Affine Algebraic Sets - D&F Chapter 15, Section 15.1 - Example 2

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