---
tag: V74WMGQQ9D
kind: concept
name: Radical extension
msc: [12F10, 12F05]
specializes: [0EQWVHF3W5]
sources:
  - url: https://en.wikipedia.org/wiki/Radical_extension
same_as:
  wikidata: Q6588186
  wikipedia: Radical extension
---

## Summary

A radical extension is a field extension built by adjoining roots one at a time.
At each step one adjoins an $n$-th root of an element already present.
Each of its elements can be written from elements of the base field with the four field operations and root extraction.
A polynomial is [[solvable by radicals|J7YS0DA3HQ]] when its roots lie in a radical extension, and [[Galois' theorem|VHCWCFA3MX]] turns this into a property of a group.

## Explanations

### L1 human

Start with the rational numbers and allow yourself to take square roots, cube roots and so on, one step at a time.
Every field you reach in this way is a radical extension of $\Q$.
For example, $\Q(\sqrt{2})$ is radical, and so is $\Q(\sqrt{2}, \sqrt[3]{1 + \sqrt{2}})$, built in two steps.
A number like $\sqrt[3]{1 + \sqrt{2}}$ is exactly the kind of expression that a formula for the roots of a polynomial produces.