Suppose that k is a field which is not algebraically closed. a. Show that if I...

Suppose that k is a field which is not algebraically closed. a. Show that if I ⊂ k[x1, . . . , xn ] is maximal, then V(I) is either empty or a point in kn . Hint: Examine the proof of Theorem 11. b. Show that there exists a maximal ideal I in k[x1, . . . , xn ] for which V(I) = ∅. Hint: See the previous exercise. c. Conclude that if k is not algebraically closed, there is always a maximal ideal of k[x1, . . . , xn ] which is not of the form <x1 − a1, . . . , xn − an >

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Colby Messinger answered 5 months ago

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Suppose that k is a field which is not algebraically closed. a. Show that if I...

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