The gap left open by guide 1
In the previous guide you built affine varieties as the common zero sets of collections of polynomials in k[x_1, ..., x_n], and you topologized affine n-space by declaring those zero sets to be the closed sets — the Zariski topology. Two maps ran in opposite directions there. From a set of polynomials S you formed its vanishing locus V(S); from a set of points X you formed the ideal I(X) of all polynomials vanishing on X. The two are visibly intertwined, but guide 1 stopped short of saying exactly how. The honest question left dangling is this: if I hand you a variety X, can you read off its defining ideal I(X) uniquely, and conversely does every ideal cut out a genuinely different variety?
The answer is no, and the failure is instructive. Over the real field, the ideal generated by x^2 + 1 has empty vanishing locus in the line — no real number squares to -1 — even though the ideal is far from everything. And the ideals (x) and (x^2) in k[x] cut out the very same point, the origin, yet they are different ideals: the second remembers a 'multiplicity' the topology cannot see. So V and I are not inverse to each other in general. The whole content of this guide is the single theorem that repairs both failures at once, provided we work over the right kind of field.
Why algebraically closed, and the weak form
The first failure — an ideal that is proper yet has no zeros — is purely a defect of the field. The fix is to work over an algebraically closed field k, meaning every non-constant polynomial in one variable has a root in k; the complex numbers C are the canonical example, and you should picture C throughout. The weak Nullstellensatz is the clean payoff: over an algebraically closed k, an ideal J in k[x_1, ..., x_n] has empty vanishing locus V(J) if and only if J is the whole ring, i.e. J = (1). Contrapositively, every proper ideal has at least one common zero. This is the right generalization of 'a degree-d polynomial has a root': systems of polynomial equations that are not algebraically forced to be inconsistent always have a solution.
It is worth being honest about why this needs proof and is not a triviality. In one variable it is just the fundamental theorem of algebra. In many variables there is no formula for a solution, and the statement is genuinely deep — one standard route uses Noether normalization and the theory of integral extensions, another uses the fact that a field finitely generated as an algebra over k must equal k (Zariski's lemma). We will not prove it here; this is a guide, not a course, and the proof is a substantial piece of commutative algebra. State the hypothesis loudly: drop 'algebraically closed' and the theorem is simply false, as x^2 + 1 over R already shows.
The strong form and the radical
The second failure — that (x) and (x^2) define the same point — is repaired by the strong Nullstellensatz, the version that earns the name 'dictionary'. It identifies exactly which polynomials vanish on the locus of an ideal. The precise statement: for any ideal J in k[x_1, ..., x_n] over an algebraically closed k, the ideal of polynomials vanishing on V(J) equals the radical of J. The radical, written rad(J), is the set of all f for which some power f^m lies in J — it is the ideal you get by 'forgetting multiplicities'. So I(V(J)) = rad(J): taking the vanishing locus and then asking what vanishes on it does not return J, it returns rad(J).
Run it on the example to feel it. The ideal (x^2) in k[x]: its radical is (x), because x is not in (x^2) but x^2 is, so x belongs to the radical. And indeed I(V((x^2))) = I({0}) = (x). The strong form thus tells you that geometry — the set of points — can only ever recover an ideal up to radical. Two ideals with the same radical are geometrically indistinguishable as point sets; the difference between (x) and (x^2) is exactly the kind of infinitesimal multiplicity data that classical varieties throw away and that schemes, the subject of a later rung, are invented to remember.
The dictionary, line by line
Now the payoff. With both forms in hand, V and I become mutually inverse, order-reversing bijections between two worlds: on the geometry side, the affine varieties (Zariski-closed subsets of affine n-space); on the algebra side, the radical ideals of k[x_1, ..., x_n] (ideals equal to their own radical). Bigger varieties correspond to smaller ideals — more points means fewer polynomials can vanish on all of them — which is why the correspondence reverses inclusions. This is the central organizing fact of the subject: every geometric question about varieties translates into an algebraic question about radical ideals, and vice versa, with no information lost.
Geometry (varieties in A^n) Algebra (radical ideals of k[x_1..x_n]) ----------------------------- ---------------------------------------- V(J) <-------------------------- J (a radical ideal) X --------------------------> I(X) I(V(J)) = rad(J) larger variety <-> smaller ideal (inclusion-reversing) empty set V(1) <-> (1) = whole ring (weak form) single point (a_1,...,a_n) <-> maximal ideal m = (x_1-a_1, ..., x_n-a_n) irreducible variety <-> prime ideal whole space A^n <-> the zero ideal (0)
Two rows of that table deserve a closer look because they refine the topology of guide 1. A variety is irreducible — not the union of two strictly smaller closed sets — exactly when its ideal is a prime ideal. And a single point corresponds to a maximal ideal; the weak Nullstellensatz says, over algebraically closed k, that every maximal ideal of k[x_1, ..., x_n] has the form (x_1 - a_1, ..., x_n - a_n), so points and maximal ideals are the same thing. The decomposition of a variety into its irreducible components then mirrors, exactly, the decomposition of its radical ideal as an intersection of primes — geometry's connected pieces are algebra's prime factors.
Cashing it in: the coordinate ring
The dictionary is not a curiosity; it is the license to attach a ring to every variety and study the variety through that ring. Given a variety X with ideal I(X), the polynomial functions on X are the polynomials in k[x_1, ..., x_n] modulo those that already vanish on X — that is, the quotient ring k[x_1, ..., x_n] / I(X). This is the coordinate ring of X, written k[X]. Because I(X) is a radical ideal (by the strong form), k[X] has no nonzero nilpotent elements: a function whose square is zero must already be zero. The Nullstellensatz is precisely what guarantees that polynomial functions on a classical variety form such a clean, reduced ring.
From here the bijection upgrades from sets to structures. The points of X correspond to the maximal ideals of k[X]; the irreducible subvarieties correspond to its prime ideals; and X is irreducible exactly when k[X] is an integral domain, in which case the function field of X — the rational functions, studied head-on in the next guide — is the fraction field of k[X]. The single integer measuring the size of X, its dimension, will be read off from k[X] too, as the length of the longest chain of primes. This is the program the whole rung executes: translate, compute in the ring, translate back.
Limits, honesty, and what comes next
Carry three honest cautions up the ladder. First, the algebraically-closed hypothesis is not negotiable for the dictionary as stated — over R or Q the maps V and I still exist and are still useful, but they stop being mutually inverse, and 'real algebraic geometry' is a genuinely different and harder subject, not a special case. Second, the strong form deliberately discards multiplicity: passing to the radical is exactly the loss of the difference between (x) and (x^2). If that infinitesimal data matters to you — and for intersection theory and deformations it does — you are being pushed toward projective varieties and ultimately schemes, where nilpotents are kept rather than killed.
Third, do not oversell the abstraction. Replacing varieties by their coordinate rings can feel like trading something vivid for something austere, but the trade earns its keep: questions about points, components, intersections, and dimensions that are slippery to picture become routine ring computations — does this polynomial lie in this radical, is this quotient a domain, how long is this chain of primes. The ring is not a mystery laid over the geometry; it is the geometry written in a language a computer or a hand can actually manipulate. That is why the category-theoretic machinery of later rungs earns its keep too, and is not abstraction for its own sake.
A clean way to remember the whole guide: over an algebraically closed field, geometry and commutative algebra are the same book in two languages, and the Nullstellensatz is the translation key — V and I are inverse bijections between varieties and radical ideals, points are maximal ideals, irreducibility is primeness, and the coordinate ring k[X] is where you actually compute. Guide 3 takes the next step, replacing global polynomial functions by regular and rational functions and the function field; guides 4 and 5 then read dimension, smoothness, and the intersection numbers of Bézout off the very same algebraic data this dictionary made available.