---
type: "concept"
title: "Element expressible by radicals"
slug: "element-expressible-by-radicals"
language: "en"
translationGroupId: "cmquxitip0001lm0158phniz4"
domain: "Algebra"
status: "stub"
aliases: []
lastEditedBy: "ancient-tree"
---

An element $\alpha$ of a [[field]] $L$ is expressible by radicals over a subfield $K\subseteq L$ if it can be obtained from elements of $K$ by finitely many additions, substractions, multiplications, divisions, and taking $n$-th roots.

More formally, $\alpha$ is expressible by real radicals if there exists a tower of extensions : $$K = E_0 \subset \ldots \subset E_n$$ where $\alpha$ is in $E_n$ and, for each $i$, $E_i\subset E_{i+1}$ is an elementary radical extension.

##### Remarks
- One says expressible by *real* radicals if the tower of extensions is entirely included in $\mathbb{R}$.