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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)

(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]

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]

(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]

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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