---
tag: HH6VEVM6Z0
kind: concept
name: Ideal of an algebraic set
msc: [14A05, 13A15]
sources:
  - url: https://en.wikipedia.org/wiki/Affine_variety
---

## 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.

## Explanations

### L1 human

Given a set of points, collect every polynomial that is zero on all of them.
These polynomials form an ideal: sums of such polynomials and multiples of them still vanish on the set.
For the $x$-axis in the plane, the ideal consists of all polynomials divisible by $y$.
For the single point $(1, 2)$, the ideal is generated by $x - 1$ and $y - 2$.