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Math Help - Affine Algebraic Sets - D&F Chapter 15, Section 15.1

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    Super Member Bernhard's Avatar
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    Affine Algebraic Sets - D&F Chapter 15, Section 15.1

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

    It follows that  I((a_1, a_2, ... ... ... a_n)) = (x - a_1, x_ a_2, ... ... ... , x - a_n)

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

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

    so in this case, clearly  g(a_1, a_2) = 0 ... ... ... and, of course, other polynomials in   \mathbb{R} [x,y] 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
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    Last edited by Bernhard; October 26th 2013 at 05:54 PM.
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