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

单词	Radical1
释义	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 $\\displaystyle\\sqrt[n]{\\frac{a}{b}}$ , where $n$ is a positive integer and $a$ and $b$ are integers with $b\eq 0$ are radicals over $\\mathbb{Q}$ . 2. The number $\\sqrt[4]{2}$ is a radical over $\\mathbb{Q}(\\sqrt{2})$ since $(\\sqrt[4]{2})^{2}=\\sqrt{2}\\in\\mathbb{Q}(\\sqrt{2})$ .
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