Let K be an algebraically closed field and let L K be any field extension. Let F₁(x₁,,n),..., Fm (1,...,n) = K[x₁,..., n

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Let K Be An Algebraically Closed Field And Let L K Be Any Field Extension Let F X N Fm 1 N K X N 1 (103.46 KiB) Viewed 36 times

Let K be an algebraically closed field and let L K be any field extension. Let F₁(x₁,,n),..., Fm (1,...,n) = K[x₁,..., n]. Use the weak form of Hilbert's Nullstellensatz to show that if the system of equations F₁(x1,...,xn) = 0 ⠀ Fm (1,,n) = 0 has a solution in I" then it has a solution in K". (Hint: if a solution exists in L", what can you say about the ideal of L1,...,n] generated by F₁,..., Fm? Also, use the previous problem.) Following is the previous problem (Only need prove above, do not need prove below) Let K be field, let R = K[x₁,...,xn] and let F₁(x1,...,n),..., Fm (1,...,n) E R. Show that Fi(k₁,..., kn) = 0 : Fm (k₁,..., kn) = 0 for (k₁,..., kn) K" if and only if the ideal I CR generated by F₁,..., Fm is contained in the maximal ideal m CR generated by 1 k₁, n - kn. Theorem 31.4 (Hilbert's Nullstellensatz weak form). Let K be an algebraically closed field, let R = K[x₁,...,n] and let fi... f. R. If f₁R++fR R, there exists a (a₁,...,an) EK" such that fi(aan) = 0 for all 1 ≤ i ≤s. Definition 35.1. Let K be a field and let R = K[₁,...,nl. For any subset XCK" we define I(X) = {fe R| f(x) = 0 for all x € X}. Proposition 35.2. Let K be a field. (a) For any UCVCK", I(U) 2 I(V). (b) I(0) = K[x₁,...,n] and if K is infinite, I(K") = 0. (c) For any collection {UAAEA of subsets of K", I(UxEAUX) = NXEA I(UX). (d) If X C K is an algebraic set, then V(I(X)) = X. Proposition 35.3. For any subset XC K", I(X) is a radical ideal. Lemma 35.4. Let K be an algebraically closed field, let R = K[₁,...,n] and let JCR be an ideal. We have I(V(J)) = √J.