The algebra-geometry dictionary: polynomial ideals correspond to geometric varieties. Master the Nullstellensatz.
At the heart of algebraic geometry lies a beautiful correspondence: polynomial equations (algebra) define geometric shapes (geometry).
This section explores ideals and varieties — the formal language that makes this correspondence precise. The Nullstellensatz is the fundamental theorem that connects them.
An ideal I is a set of polynomials. Its variety V(I) is all points where every polynomial in I equals zero. Explore the connection.
An ideal I ⊆ k[x,y] is a set of polynomials closed under addition and multiplication by any polynomial. The variety V(I) is the set of all points where every polynomial in I vanishes. Generators are polynomials that "span" the ideal.
Given a set of points (a variety), which polynomials vanish on them? Click to add points and discover the ideal of the variety.
Click to add points (up to 5). Click a point to remove it. The ideal I(V) contains all polynomials vanishing on your points.
Given an ideal I, the variety V(I) is all points where every polynomial in I vanishes.
Given a variety V, the ideal I(V) is all polynomials that vanish on every point of V.
Hilbert's Nullstellensatz is the fundamental theorem of algebraic geometry. It establishes when the ideal-variety correspondence is a perfect bijection.
A maximal ideal corresponds to a single point. This ideal is radical.
The Nullstellensatz establishes a bijection between:
This is the foundation of algebraic geometry: algebra ⟷ geometry
Algebraic operations on ideals correspond to geometric operations on varieties. Explore how sums, products, and intersections of ideals translate geometrically.
Notice: Sums become intersections, products/intersections become unions!
You've learned the fundamental language of algebraic geometry:
This algebra-geometry dictionary is why the field is called "algebraic" geometry. Every geometric question can be translated to algebra, and vice versa!