If \(F\) is a radical extension field of \(K\) and \(E\) is an intermediate field, then \(F\) is a radical extension of \(E\).
Solution
Since \(F/K\) is radical, \(F=K(\alpha_1,\ldots,\alpha_n)\), where for each \(i\) there is \(n_i>0\) with \(\alpha_i^{n_i}\in K(\alpha_1,\ldots,\alpha_{i-1})\).
1. \(F=E(\alpha_1,\ldots,\alpha_n)\).
Since \(K\subseteq E\subseteq F\), we have \(F=K(\alpha_1,\ldots,\alpha_n)\subseteq E(\alpha_1,\ldots,\alpha_n)\subseteq F\).
2. For each \(i\), \(\alpha_i^{n_i}\in E(\alpha_1,\ldots,\alpha_{i-1})\).
\(\alpha_i^{n_i}\in K(\alpha_1,\ldots,\alpha_{i-1})\subseteq E(\alpha_1,\ldots,\alpha_{i-1})\) because \(K\subseteq E\).
3. Q.E.D.
Steps 1 and 2 exhibit \(F\) as \(E(\alpha_1,\ldots,\alpha_n)\) with \(\alpha_i^{n_i}\in E(\alpha_1,\ldots,\alpha_{i-1})\) for every \(i\), which is the definition of a radical extension of \(E\).