An algebraic variety V(f₁,...,fₖ) is the set of common zeros of a collection of polynomials — the fundamental geometric objects of algebraic geometry. Varieties can be irreducible (not decomposable as a union of smaller varieties) or decompose into irreducible components. The Nullstellensatz (Hilbert 1893) gives the precise correspondence between radical ideals and affine varieties over algebraically closed fields: I(V(J)) = √J. Dimension, singularities, and topology of varieties are key invariants: an elliptic curve is a smooth genus-1 curve with a group structure, while a singular point is where all partial derivatives vanish simultaneously. Wiles' proof of Fermat's Last Theorem ultimately reduced to properties of elliptic curves over rational numbers.