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**demode** So since the dimension is 2, there are 2 elements in its basis. But the question doesn't ask how many there are, it's asking to identify them! Do I somehow need to find two elements in $\displaystyle \mathbb{Q}$ such that $\displaystyle \mathbb{Q}(\sqrt{7})$ is spanned by them?

So I must show that generators of $\displaystyle \mathbb{Q}(\sqrt{-3})$ and $\displaystyle \mathbb{Q}(\sqrt{5})$ are not the roots of the same polynomial. How do I find them? I'm very confused...

Here's an alternative approach:

Suppose that $\displaystyle \phi: \mathbb{Q}(\sqrt{-3}) \to \mathbb{Q}(\sqrt{5})$ is an isomorphism. Since $\displaystyle \phi(1) = 1$ (is this right?), we have $\displaystyle \phi(-3)=-3$. Then

$\displaystyle -3=\phi(-3)=\phi(\sqrt{-3} \sqrt{-3})= (\phi(\sqrt{-3}))^2$

This is impossible, since $\displaystyle \phi(\sqrt{-3})$ is a real number.

Is this correct?