So I'm considering a Field $\mathbb{F}$, such that $\mathbb{Q}$ is a subset of $\mathbb{F}$ and when it's considered a vector space over $\mathbb{Q}$, it has dimension 2. I want to show two things:
1) There exists an element a of $\mathbb{F}$ that's not in $\mathbb{Q}$ such that it satisfies the equation $a^2-n=0$ for some $n\in\mathbb{Z}$.
2) That $\mathbb{F}$ is isomorphic to $\mathbb{Q}[\surd(n)]$ and further that $n$ is square-free.
So, showing there's an element in the complement of $\mathbb{F}$ and $\mathbb{Q}$ I've managed, but I can't show that it satisfies the equation in question. For number 2, I'm lost.
(Trying LaTeX, hope I didn't screw it up too bad.)