Find the primitive ideal for a given Galois extension.

Question

Could $E=\{a_0*e+a_1*x+a_2*x^2|a_0,a_1,a_2∈\mathbb Q\}$ $(x^3=e)$be an extention field for $\mathbb Q$? If yes, what could the corresponding $f(x)∈\mathbb Q[x]$ be for $E\cong \mathbb Q[x]/<f(x)>$?

Answer 1

Talking about extension fields of $\mathbb{Q}[x]$ doesn't really make sense since $\mathbb{Q}[x]$ is not a field, but I assume you meant extensions of $\mathbb{Q}$. Also since $x^3=1$ in $E$, we see that $E \cong \mathbb{Q}[x] / \langle x^3-1 \rangle$ which is not a field since $x^3-1$ is reducible.