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 $\displaystyle (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 $\displaystyle k[x_1, x_2, ... ... x_n] $ to the field k given by evaluation at $\displaystyle (a_1, a_2, ... ... ... a_n) $.

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

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

$\displaystyle 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 $\displaystyle g(a_1, a_2) = 0 $ ... ... ... and, of course, other polynomials in $\displaystyle \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