Hilbert's Nullstellensatz and radical ideals

The weak Nullstellensatz identifies each maximal ideal of $k[x_1,\ldots,x_n]$ with $(x_1-a_1,\ldots,x_n-a_n)$ for one point a when k is algebraically closed. Equivalently every proper ideal has a common k-valued zero. The finite generation condition matters: the proof uses Zariski's lemma, which says a field finitely generated as a k-algebra is finite algebraic over k. Algebraic closedness then collapses that residue-field extension to k itself.

The strong form says that the ideal of all polynomials vanishing on V(J) is the radical of J. Neither characteristic zero nor perfection needs to be added: algebraically closed fields in every characteristic satisfy the theorem. Over the reals, the proper ideal generated by x²+1 has no real zero, so the same statement about real-valued points fails. Over a finite field with q elements, x^q−x vanishes at every field element despite being a nonzero polynomial. These are field failures, not exceptions to the algebraically closed theorem.

A useful proof of the strong form starts with a polynomial f that vanishes on V(J). In a polynomial ring with one extra variable t, consider J together with 1−tf. A common zero would require f to be both zero and invertible, which is impossible. By the weak form the enlarged ideal is the whole ring. An identity expressing one as a combination of these generators therefore exists, even though it need not be easy to find computationally.

Substitute t=1/f in the localization where f is invertible. The generator 1−tf disappears, leaving an expression for one in the localized extension of J. Clearing finitely many powers of f from denominators gives f^r in J. This Rabinowitsch argument proves radical membership rather than membership of f itself. It separates an existence theorem from an efficient certificate algorithm: the theorem guarantees a finite power and finite combination, but does not promise that either has a small degree or is numerically well conditioned.

The radical quotient k[x]/sqrt(J) is reduced, whereas the original quotient k[x]/J may contain nonzero nilpotent classes. Its ordinary points agree with those of the reduced quotient. Nilpotence is useful information about infinitesimal thickening and multiplicity, so taking the radical is not an innocuous algebraic simplification in every problem. It is appropriate when the object under study is an ordinary algebraic set, but may erase the feature a scheme or intersection calculation was intended to preserve.

For a monomial ideal, surviving monomials give an especially transparent way to test these distinctions. A polynomial class vanishes in the quotient precisely when all its surviving coefficients vanish. This can establish that one power belongs to the ideal and a smaller power does not. Exponents and coefficients both matter: in positive characteristic a binomial coefficient can disappear, changing the least nilpotence exponent of a particular sum even though its radical membership remains unchanged. We therefore state characteristic zero in the explicit calculation below.