---
tag: BSDCDJR6RQ
kind: concept
name: Bring radical
msc: ["12E12", "33E20"]
sources:
  - url: https://en.wikipedia.org/wiki/Bring_radical
same_as:
  wikidata: Q2386216
  wikipedia: Bring radical
---

## Summary

The Bring radical of a real number $a$ is the unique real root of $x^5 + x + a$.
It plays for quintic equations the role that $n$-th roots play for the equations $x^n = a$.
Using only radicals, every quintic equation can be reduced to the Bring–Jerrard form $x^5 + px + q = 0$.
With the Bring radical extended to complex arguments, the roots of every quintic can then be written with radicals and Bring radicals.

## Explanations

### L1 human

The [[Abel–Ruffini theorem|163A2WQQJX]] says that ordinary roots are not enough to solve all quintic equations.
The Bring radical is a new kind of root that fills this gap.
Just as $\sqrt[3]{a}$ is the real number whose cube is $a$, the Bring radical of $a$ is the real number $x$ with $x^5 + x = -a$.
For example, the Bring radical of $-2$ is $1$, since $1^5 + 1 - 2 = 0$.