This is just a quick question about field extensions. It does have important implications for a problem that I am currently working on, so even a simple answer would be useful. I understand that, by definition, $\mathbb{Q}(\sqrt2,\sqrt{1+i})$ is the smallest subfield of $\mathbb{C}$ containing $\mathbb{Q}$, $\sqrt2$ and $\sqrt{1+i}$. Is $\mathbb{Q}_\sqrt2(\sqrt{1+i}) = \mathbb{Q}(\sqrt2,\sqrt{1+i})$?
That is; is adjoining $\sqrt{1+i}$ to the base field $\mathbb{Q}(\sqrt2)$ necessarily the same as the smallest field containing those elements, i.e. $\mathbb{Q}(\sqrt2,\sqrt{1+i})$?