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- Jun 22, 2012

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Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows:

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If [TEX] k = \mathbb{F}_2 [/TEX] and [TEX] V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 [/TEX],

show that [TEX] \mathcal{I} (V) [/TEX] is the product ideal [TEX] m_1m_2 [/TEX]

where [TEX] m_1 = (x,y) [/TEX] and [TEX] m_2 = (x -1, y-1) [/TEX].

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I am having trouble getting started on this problem.

One issue/problem I have is - what is the exact nature of [TEX] m_1, m_2 [/TEX] and [TEX] m_1m_2 [/TEX]. What (explicitly) are the nature of the elements of these ideals.

I would appreciate some help and guidance.

Peter

Note: D&F define [TEX] \mathcal{I} (V) [/TEX] as follows:

\(\displaystyle \mathcal{I} (V) = \{ f \in k(x_1, x_2, ......... , x_n) \ | \ f(a_1, a_2, ......... , a_n) = 0 \ \ \forall \ \ (a_1, a_2, ......... , a_n) \in V \} \)

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If [TEX] k = \mathbb{F}_2 [/TEX] and [TEX] V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 [/TEX],

show that [TEX] \mathcal{I} (V) [/TEX] is the product ideal [TEX] m_1m_2 [/TEX]

where [TEX] m_1 = (x,y) [/TEX] and [TEX] m_2 = (x -1, y-1) [/TEX].

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

I am having trouble getting started on this problem.

One issue/problem I have is - what is the exact nature of [TEX] m_1, m_2 [/TEX] and [TEX] m_1m_2 [/TEX]. What (explicitly) are the nature of the elements of these ideals.

I would appreciate some help and guidance.

Peter

Note: D&F define [TEX] \mathcal{I} (V) [/TEX] as follows:

\(\displaystyle \mathcal{I} (V) = \{ f \in k(x_1, x_2, ......... , x_n) \ | \ f(a_1, a_2, ......... , a_n) = 0 \ \ \forall \ \ (a_1, a_2, ......... , a_n) \in V \} \)

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