I'm wondering if my thinking here is right. I want to find all subfields of $\mathbb{Q}$($\sqrt2,\sqrt3)$. I believe that $\mathbb{Q}$($\sqrt2,\sqrt3)$ is the smallest field containing $\mathbb{Q}$, $\sqrt3$, and $\sqrt2$.
So this leads me to believe all of the subfields are: $\mathbb{Q}$($\sqrt2,\sqrt3)$ (the field itself), $\mathbb{Q}$($\sqrt2)$,$\mathbb{Q}$($\sqrt3)$, and $\mathbb{Q}$. Is this right?
