How to prove that $\sqrt{2-\sqrt{2}} \in \mathbb{Q}(\sqrt{2+\sqrt{2}})$ I am trying to prove a statement about the decomposition field of a polynomial that has both $\sqrt{2-\sqrt{2}}$ and $\sqrt{2+\sqrt{2}}$ as roots. I cannot find a way to prove that $\sqrt{2-\sqrt{2}} \in \mathbb{Q}(\sqrt{2+\sqrt{2}})$. I have tried writing it in the basis $1,\sqrt{2+\sqrt{2}}(\sqrt{2+\sqrt{2}})^2,(\sqrt{2+\sqrt{2}})^3$ but nothing works and without this I cannot prove that the decomposition field of $t^4-4t^2+2$ is $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ over $\mathbb{Q}$

Using the hint given by Bart Michels, we have$$
\sqrt{2-\sqrt{2}}\sqrt{2+\sqrt{2}}=\sqrt{2}\\
\sqrt{2+\sqrt{2}}(\sqrt{2-\sqrt{2}})-\sqrt{2}=0
$$
And so the minimial polynomial of $\sqrt{2-\sqrt{2}}$ is linear in $\mathbb{Q}(\sqrt{2+\sqrt{2}})$ and so it must be an element of the field.
