$Μ : K$ need not be radical Show that $Μ : K$ need not be radical, Where $L : K$ is a radical extension in $ℂ$ and $Μ$ is an intermediate field. 

Let $K = \Bbb Q$ and $M$ be the splitting field of $X^3 - 3X + 1 \in \Bbb Q[X]$.
$M$ can be embedded into $\Bbb R$, so it is not a radical extension by casus irreducibilis.
However, $X^3 - 3X + 1$ has a solvable Galois group $C_3$ (the cyclic group of order $3$), so $M$ can be embedded into some field $L$ that is radical over $K = \Bbb Q$.
