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 ${(\sqrt[4]{2})}^2=\sqrt{2}\in \mathbb{Q}(\sqrt{2})$.