---
tag: J7YS0DA3HQ
kind: concept
name: Solvable by radicals
msc: [12F10, "12E12"]
sources:
  - url: https://en.wikipedia.org/wiki/Solution_in_radicals
---

## Summary

A polynomial is solvable by radicals when all its roots can be written from its coefficients with the four field operations and $n$-th roots.
Formally, its splitting field lies in a [[radical extension|V74WMGQQ9D]] of the base field.
In characteristic $0$, [[Galois' theorem|VHCWCFA3MX]] says this holds exactly when the [[Galois group of the polynomial|0JZY3R908H]] is [[solvable|0P3KZK0FTV]].

## Explanations

### L1 human

The quadratic formula shows that every equation $ax^2 + bx + c = 0$ is solvable by radicals: its roots are $(-b \pm \sqrt{b^2 - 4ac})/(2a)$.
The [[cubic formula|FB6J5WAT5V]] and the [[quartic formula|4CDFMKKZ01]] do the same for degrees $3$ and $4$.
The [[Abel–Ruffini theorem|163A2WQQJX]] says that no such formula exists for the general equation of degree $5$.
Some quintics are still solvable by radicals, such as $x^5 - 2 = 0$, whose roots are $\sqrt[5]{2}$ times the fifth roots of unity.