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coordinate ring questions

Hello!

I need your help to solve this question

A subset X⊂k^n (k a field) is called algebraic if there exist polynomials f1,…,fm∈k[t1,…,tn] such that

X={x∈Kn|f1(x)=...=fm(x)=0}. The coordinate ring k[X] of X is the ring of all functions f:X→k that can be represented by some polynomial. That is, there exists a polynomial g∈k[t1,…,tn] such that f(x)=g(x) for all x∈X.

1- Show that for two different points x,y∈X there exists f∈k[X] with f(x)≠f(y).

2- Let k=Z_2=Z/2Z and X⊂k^2 be the set {(0,1),(1,0)}. Is X algebraic? Determine its coordinate ring.

3- Let g_i in N be a sequence of polynomials and define Y:={x∈k^n|gi(x)=0 ∀ i}. Show that Y is algebraic set.

Thanks

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