Subrings between $\mathbb{Q}$ and $\mathbb{Q}[\sqrt{2}]$.

Question

I want to find all the intermediate subrings between $\mathbb{Q}$ and $\mathbb{Q}[\sqrt{2}]$.

Thank you.

Answer 1

In general, any intermediate ring of an algebraic field extension is a field. This is a very easy exercise. Thus it boils down to intermediate fields and probably you can solve this problem yourself.

But one should note that one can easily solve the problem without that knowledge: Regardless of the fact whether a subring is a field or not, it is definitely a $\mathbb Q$-vector space and as such it is a subspace of the $2$-dimensional space $\mathbb Q(\sqrt 2)$. I think you can carry on from here.