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Let $S=\{s+t\sqrt2+(u+v\sqrt2)i\mid s,t,u,v\in\mathbb Q\}$. Find all subfields of $S$.

Clearly S is a subfield of S. Past that I'm not really sure where to start. The textbook doesn't give many notes and my professor doesn't lecture.

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    $S=(\mathbb Q{\sqrt2})[i]=\mathbb Q(\sqrt2,i)$ – 2017-02-08

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Hint: Note that $S={\bf Q}[\sqrt 2,i]$. It follows that the Galois group of $S$ over ${\bf Q}$ is isomorphic to $({\bf Z}/2{\bf Z})^2$ (check it!), so it's fairly easy to find all its subgroups. Apply the Galois theorem to find all subfields of $S$.