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

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If $\displaystyle k = \mathbb{F}_2 $ and $\displaystyle V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2 $,

show that $\displaystyle \mathcal{I} (V) $ is the product ideal $\displaystyle m_1m_2 $

where $\displaystyle m_1 = (x,y) $ and $\displaystyle m_2 = (x -1, y-1) $.

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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 $\displaystyle m_1, m_2 $ and $\displaystyle m_1m_2 $. What (explicitly) are the nature of the elements of these ideals.

I would appreciate some help and guidance.

Peter

Note: D&F define $\displaystyle \mathcal{I} (V) $ as follows:

$\displaystyle \mathcal{I} (V) = \{ f \in k(x_1, x-2, ......... , x_n) \ | \ f(a_1, a_2, ......... , a_n) = 0 $ for all $\displaystyle (a_1, a_2, ......... , a_n) \in V \} $