Is $\mathbb Q[\sqrt{11} + 10^\frac{1}{3}]$ equal to $\mathbb Q[\sqrt{11}, 10^\frac{1}{3}]$?

Question

Is $\mathbb Q[\sqrt{11} + 10^\frac{1}{3}]$ equal to $\mathbb Q[\sqrt{11}, 10^\frac{1}{3}]$?

It is clear that $\sqrt{11} + 10^\frac{1}{3} \in \mathbb Q[\sqrt{11}, 10^\frac{1}{3}]$ and that $\mathbb Q \subset \mathbb Q[\sqrt{11}, 10^\frac{1}{3}]$. Thus $\mathbb Q[\sqrt{11} + 10^\frac{1}{3}] \subset \mathbb Q[\sqrt{11}, 10^\frac{1}{3}]$.

Can we somehow prove that either $\sqrt{11}$ or $10^\frac{1}{3} \in \mathbb Q[\sqrt{11} + 10^\frac{1}{3}]$ and demonstrate that $\mathbb Q[\sqrt{11}, 10^\frac{1}{3}] \subset \mathbb Q[\sqrt{11} + 10^\frac{1}{3}]$.

Thanks!

Answer 1

Let $\newcommand{\al}{\alpha}\al=\sqrt{11}+10^{1/3}$. Then $$(\al-\sqrt{11})^3=10.$$ So $$\al^3+33\al-10=(3\al^2+11)\sqrt{11}$$ and then $\sqrt{11}\in\Bbb Q(\al)$.