radicalRadical52007-04-14 21:10:492007-04-21 20:28:34Definition\usepackage{amssymb}
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Let $F$ be a field and $\alpha$ be \PMlinkname{algebraic}{Algebraic} over $F$. Then $\alpha$ is a \emph{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:
\begin{enumerate}
\item 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 $\mathbb{Q}$.
\item The number $\sqrt[4]{2}$ is a radical over $\mathbb{Q}(\sqrt{2})$ since $(\sqrt[4]{2})^2=\sqrt{2} \in \mathbb{Q}(\sqrt{2})$.
\end{enumerate}