PlanetMath: radical

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radical

(Definition)

Let be a field and $\alpha$ be algebraic over . Then $\alpha$ is a radical over if there exists a positive integer with $\alpha^n \in F$ .

Note that, if is a field extension and $\alpha$ is a radical over , then $\alpha$ is automatically a radical over .

Following are some examples of radicals:

All numbers of the form $\displaystyle \sqrt[n]{\frac{a}{b}}$ , where is a positive integer and and are integers with $b \neq 0$ are radicals over $\mathbb{Q}$ .
The number $\sqrt[4]{2}$ is a radical over $\mathbb{Q}(\sqrt{2})$ since $(\sqrt[4]{2})^2=\sqrt{2} \in \mathbb{Q}(\sqrt{2})$ .

"radical" is owned by Wkbj79.

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Cross-references: field extension, integer, positive, field
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This is version 6 of radical, born on 2007-04-14, modified 2007-04-21.
Object id is 9190, canonical name is Radical5.
Accessed 761 times total.

Classification:

AMS MSC:	12F10 (Field theory and polynomials :: Field extensions :: Separable extensions, Galois theory)
	12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions)

Pending Errata and Addenda

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