Ideal of an algebraic set

Summary

The ideal of a subset $X$ of affine space $k^n$ is the set of all polynomials that vanish at every point of $X$. It is the algebraic shadow of $X$ and is always a radical ideal. Over an algebraically closed field, Hilbert's Nullstellensatz says that this gives a bijection from algebraic sets to radical ideals. This turns geometry into algebra.