Proving an element belongs to field extension

Question

I am unsure of questions asking to prove that an element belongs to a field extension. Here is an example:

Prove that $\sqrt2 \in \mathbb{Q}( \sqrt{11+3\sqrt{13} } )$

$\sqrt2 \notin \mathbb{Q}$

Let:

$x=\sqrt{11+3\sqrt{13}} \implies (\frac{x^2-11}{3})^2-13=0 \implies x^4-22x^2+82=0$

Let:

$x^2=y \implies y^2-22y+82=0 \implies y= \frac{22 \pm \sqrt{22^2-4\times82}}{2}=11 \pm \sqrt{11^2-82}=11 \pm \sqrt{39}$

So it appears that $\sqrt2 \notin \mathbb{Q}( \sqrt{11+3\sqrt{13} } )$, is this correct?

Answer 1

Hint $$\sqrt{11+3\sqrt{13}}=\frac{3+\sqrt{13}}{\sqrt{2}}.$$