radical
Let $F$ be a field and $\alpha $ be algebraic (http://planetmath.org/Algebraic) over $F$. Then $\alpha $ is a radical^{} over $F$ if there exists a positive integer $n$ with ${\alpha}^{n}\in F$.
Note that, if $K/F$ is a field extension and $\alpha $ is a radical over $F$, then $\alpha $ is automatically a radical over $K$.
Following are some examples of radicals:

1.
All numbers of the form $\sqrt[n]{{\displaystyle \frac{a}{b}}}$, where $n$ is a positive integer and $a$ and $b$ are integers with $b\ne 0$ are radicals over $\mathbb{Q}$.

2.
The number $\sqrt[4]{2}$ is a radical over $\mathbb{Q}(\sqrt{2})$ since ${(\sqrt[4]{2})}^{2}=\sqrt{2}\in \mathbb{Q}(\sqrt{2})$.
Title  radical 

Canonical name  Radical1 
Date of creation  20130322 16:55:36 
Last modified on  20130322 16:55:36 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  9 
Author  Wkbj79 (1863) 
Entry type  Definition 
Classification  msc 12F05 
Classification  msc 12F10 
Related topic  RadicalExtension 
Related topic  NthRoot 
Related topic  SolvableByRadicals 
Related topic  Radical6 