If $\mathbb{Q}(\zeta)=\mathbb{Q}(\eta), [\mathbb{Q}(f(\zeta)):\mathbb(Q)]=[\mathbb{Q}(f(\eta)):\mathbb(Q)]$ must $\mathbb(Q)(\zeta)=\mathbb(Q)(\eta)$?

Question

Let $f$ be a polynomial with integer coefficients. Let $\mathbb{Q}(\zeta)=\mathbb{Q}(\eta)$ be an algebraic extension. If $\mathbb{Q}(\zeta)=\mathbb{Q}(\eta)$ and $[\mathbb{Q}(f(\zeta)):\mathbb(Q)]=[\mathbb{Q}(f(\eta)):\mathbb(Q)]$, must $\mathbb(Q)(\zeta)=\mathbb(Q)(\eta)$?

(Note: I used $\zeta$ and $\eta$ because if I used $\alpha$ and $\beta$ the title would be too long.)

Answer 1

No; for a counterexample, let $\zeta=\sqrt{2}+\sqrt{3}$, $\eta=\sqrt{3}+\sqrt{6}$, and $f(x)=x^2$.