Let $F$ be a field and $\alpha$ be 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 $\displaystyle \sqrt[n]{\frac{a}{b}}$ where $n$ is a positive integer and $a$ and $b$ are integers with $b \neq 0$ are radicals over $\mathbb{Q}$
2. The number $\sqrt[4]{2}$ is a radical over $\mathbb{Q}(\sqrt{2})$ since $(\sqrt[4]{2})^2=\sqrt{2} \in \mathbb{Q}(\sqrt{2})$