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Homeradical

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# radical

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

Related:

RadicalExtension, NthRoot, SolvableByRadicals, Radical6

Type of Math Object:

Definition

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Reference

## Mathematics Subject Classification

12F05*no label found*12F10

*no label found*

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