Example of a finite field extension

Question

I have this problem "Give an example about a field extension $F/K$ s.t. $|F:K|=3$ but $F\neq K(\sqrt[3]x)$ $\forall x\in K$".

I think I have a solution. Let $K=\mathbb{Q}$ and $F=\mathbb{Q}(a)$ in which $a^3+a+1=0$, then I suppose there's some $x$ s.t. $\mathbb{Q}(\sqrt[3]x)=\mathbb{Q}(a)$, then $\sqrt[3]x=\frac{x_1a^2+x_2a+x_3}{x_4}$ in which $x_1,...,x_4\in \mathbb{Z}$. Then I solve it and lead to contradiction.

What I want to ask here is that do we have a more simple solution? Thank you

Answer 1

$\mathbb{F}_8$ is degree 3 over $\mathbb{F}_2$, and is not the extension field of the cube roots of $1$, the only candidate. There is no smaller example!