Hilbert's Nullstellensatz
/ NULL-shtell-en-zatz /
There is a translation between algebra (ideals of polynomials) and geometry (the shapes they cut out). The Nullstellensatz is the dictionary that makes this translation faithful. In one direction you take an ideal of polynomials and form its zero set; in the other you take a geometric set and collect all polynomials vanishing on it. The theorem tells you exactly how far these two operations are from being inverse to each other — and the only obstruction is the difference between an ideal and its radical.
Work over an algebraically closed field k. The weak Nullstellensatz says: if an ideal I in k[x_1, ..., x_n] is proper (not the whole ring), then its zero set V(I) is nonempty. Equivalently, the maximal ideals of k[x_1, ..., x_n] are exactly the (x_1 - a_1, ..., x_n - a_n) for points a in A^n — there are no exotic maximal ideals hiding. The strong Nullstellensatz upgrades this: I(V(I)) equals the radical of I, where the radical sqrt(I) = { f : f^m is in I for some m >= 1 } collects all polynomials some power of which lands in I. So the maps V and I set up a perfect, order-reversing bijection between radical ideals and algebraic sets, and refine to a bijection between prime ideals and irreducible varieties, and between maximal ideals and points.
This is the foundational theorem of classical algebraic geometry: it is what licenses thinking about a variety entirely through its coordinate ring, and it underlies dimension theory and the prime-correspondence. Two honest caveats. First, algebraic closure is essential: over R the ideal (x^2 + 1) is proper but has empty zero set, breaking the weak form. Second, the radical is unavoidable: (x^2) and (x) have the same zero set {0} in A^1, so V cannot distinguish them, and only after passing to schemes (which remember the nilpotent x^2) does the non-reduced information survive — the Nullstellensatz is precisely a variety-level statement.
In k[x] take I = (x^2 - 2x + 1) = ((x-1)^2). Its zero set is the single point {1}, and I(V(I)) = (x - 1) = sqrt(I): the radical strips off the squaring. So the radical ideal (x - 1) and the geometric point 1 correspond perfectly, but the ideal (x-1)^2 itself does not — the lost square is exactly the non-reduced data that schemes recover.
V and I are inverse bijections only after passing to radical ideals; the radical is the exact price of working with varieties rather than schemes.
The Nullstellensatz fails over non-algebraically-closed fields and says nothing about non-reduced structure: it is a statement about varieties, not schemes. Do not invoke it where nilpotents matter.