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- Jun 22, 2012

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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)

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

It follows that [TEX] I((a_1, a_2, ... ... ... a_n)) = (x - a_1, x - a_2, ... ... ... , x - a_n) [/TEX]

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

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

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

Note - see attachment for definition of I(A)

[Note: This has also been posted on MHF]

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

It follows that [TEX] I((a_1, a_2, ... ... ... a_n)) = (x - a_1, x - a_2, ... ... ... , x - a_n) [/TEX]

-------------------------------------------------------------------------------------------------------------------------------------

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

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

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

Note - see attachment for definition of I(A)

[Note: This has also been posted on MHF]

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