Find an extension $F/L$ such that $F/\mathbb{Q}$ is radical Let $f(x)= x^3 - 7x +7 \in \Bbb Q[x]$ and let $L/\Bbb Q$ be the splitting field of $f$.
I have shown that the Galois group of $L/\Bbb Q$ is $C_3 $ and so as this is solvable, $f$ is solvable by radicals. 
I have also shown that the splitting field $L/\Bbb Q$ is not radical. 
I now need to find an extension $F/L$ such that $F/\Bbb Q$ is radical. I'm really not sure how to approach this given I don't explicitly know the roots of $f$. The question hints that I may use standard facts about Kummer extensions, but I'm not sure how I can use Kummer extensions to help. 
Any help would be greatly appreciated! :) 

The discriminant is a square and the polynomial is irreducible (Eisenstein at $7$) thus $L=\Bbb{Q}(a)$ where $a$ is a root. Let $K=\Bbb{Q}(\zeta_3)$, then $K(a)/K$ is Galois of degree $3$ and $Gal(K(a)/K)$ is cyclic of order $3$, generated by $g$. So $b=a+\zeta_3 g(a)+\zeta_3^2 g^2(a)$ satisfies $g(b)=\zeta_3^{-1}b$ and $b^3\in K$.
