In Dummit and Foote Chapter 15, Section 15.3: Radicals and Affine Varieties on page 679 we find the following definition of affine variety: (see attachment)(adsbygoogle = window.adsbygoogle || []).push({});

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Definition. A nonempty affine algebraic set [itex] V [/itex] is called irreducible if it cannot be written as [itex] V = V_1 \cup V_2 [/itex] where [itex] V_1 [/itex] and [itex] V_2 [/itex] are proper algebraic sets in [itex] V [/itex].

An irreducible affine algebraic set is called an affine variety.

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Dummit and Foote then prove the following results:

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Proposition 17. The affine algebraic set [itex] V [/itex] is irreducible if and only if [itex] \mathcal{I}(V) [/itex] is a prime ideal.

Corollary 18. The affine algebraic set [itex] V [/itex] is a variety if and only if its coordinate ring [itex] k[V] [/itex] is an integral domain.

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Then in Example 1 on page 681 (see attachment) D&F write:

"Single points in [itex] \mathbb{A}^n [/itex] are affine varieties since their corresponding ideals in [itex] k[A^n] [/itex] are maximal ideals."

I do not follow this reasoning.

Can someone please explain why the fact that ideals in [itex] k[A^n] [/itex] that correspond to single points are maximal

imply that single points in [itex] A^n [/itex] are affine varieties.

Presumably Proposition 17 and Corollary 18 are involved but I cannot see the link.

I would appreciate some help.

Peter

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# Affine Varieties - Single Points and maximal ideals

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