Radical extension

Remarks

Example: $L = \Q(\sqrt{2}, \sqrt[3]{1 + \sqrt{2}})$ is radical over $\Q$, with $\alpha_1 = \sqrt{2}$, $n_1 = 2$, $\alpha_2 = \sqrt[3]{1 + \sqrt{2}}$, $n_2 = 3$, and $\alpha_2^3 = 1 + \sqrt{2} \in \Q(\alpha_1)$. Non-example: let $\theta = \cos(2\pi/7)$, a root of the irreducible cubic $8x^3 + 4x^2 - 4x - 1$. The field $\Q(\theta)$ is real, and its three conjugates $\cos(2\pi k/7)$, $k = 1, 2, 3$, are real, so by the casus irreducibilis $\theta$ lies in no radical extension of $\Q$ contained in $\R$. In particular $\Q(\theta)/\Q$ is not radical, although $\Q(\theta)$ lies inside the radical extension $\Q(\zeta_7)$, with $\zeta_7 = e^{2\pi i/7}$. A radical extension is finite, but it need not be normal or separable: $\Q(\sqrt[3]{2})/\Q$ is radical and not normal.