Examples in finite extensions.

Question

Give an example of a field extension $K/F$ with $[K:F]=3$ but with $K \neq F(\sqrt[3]{b})$ for any $b \in F.$

I think in $F=\mathbb{Q}, K=F(w)$ with $w=e^{\frac{2\pi i}{3}}$. But I don't know if it's correct.

Any hint? Thanks for the advance!

Answer 1

Take $\;F=\Bbb F_2\;,\;\;K=\Bbb F_2[x]/\langle x^3+x+1\rangle\;$ . Then, $\;K=F(\omega)\;$ , with $\;\omega\neq\sqrt[3]\alpha\;$ , for no $\;\alpha\in F\;$

Answer 2

Consider $x^3+x+1 \in\mathbb{F_2}[x]$ it is irreducible so the quotient $$\frac{\mathbb{F_2}[x]}{(x^3+x+1)}$$ is a field (because is the quotient of a ring by the ideal generated by an irreducible polinomial so the ideal is maximal) extention of $\mathbb{F}_2$ of degree 3 that satisfy your condition.