Thread: Elementary Algebraic Geometry - D&F Section 15.1 - Exercise 15

1. Elementary Algebraic Geometry - D&F Section 15.1 - Exercise 15

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

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

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

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

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

I would appreciate some help and guidance.

Peter

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

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