---
tag: FMPJJBVM5A
kind: concept
name: Symbolic power
msc: ["13A15"]
sources:
  - url: https://en.wikipedia.org/wiki/Symbolic_power_of_an_ideal
same_as:
  wikidata: Q18393178
  wikipedia: Symbolic power of an ideal
---

## Summary

Let $P$ be a prime ideal and $n \ge 1$.
The $n$-th symbolic power $P^{(n)}$ is the set of ring elements that land in $P^n$ after multiplication by some element outside $P$.
It is the smallest [[primary ideal|TMTPWVETQK]] with radical $P$ that contains $P^n$, so it corrects the fact that $P^n$ itself need not be primary.
Symbolic powers appear in the proof of [[Krull's principal ideal theorem|SJZW2HN1SC]].
Over an algebraically closed field, they give the polynomials that vanish to order at least $n$ along a variety.

## Explanations

### L1 human

The ordinary power $P^n$ can contain extra junk located at points other than the generic point of $P$.
The symbolic power throws in everything that belongs to $P^n$ "near" $P$, and so keeps only the part of $P^n$ at $P$.
For a prime generated by some of the variables of a polynomial ring, such as $(x, y)$ in $k[x, y, z]$, nothing changes and $P^{(n)} = P^n$.
For the cone $xy = z^2$ and $P = (x, z)$, the element $x$ lies in $P^{(2)}$ but not in $P^2$.