Let $K$ be a number field and let $O$ be a subring of the ring of integers $O_K$ of $K$. Show that $O$ contains a $\mathbb{Q}$-basis of $K$ if and only if the field of fractions of $O$ is $K$.
I proved the "only if" part.
Now for the "if" part I have a hint: Show that it makes sense to talk about the largest subfield $K_0$ of $K$ such that $O$ contains a $\mathbb{Q}$-basis for $K_0$. Then assume for contradiction that $K_0 \neq K$.
Clearly $\mathbb{Q}$ is a subfield of $K$ and $O$ contains a $\mathbb{Q}$-basis of it (namely $1$). So it has sense to talk about the largest subfield of $K$ satisfying this condition. I have two main ideas:
$1$) Since $K_0 \neq K$ there exists $\alpha \in K-K_0$. I can write $\alpha=\frac{\beta}{\gamma}$ for some $\beta, \gamma \neq 0 \in O$ since $K$ is the field of fractions of $O$. Now I would like to write $\alpha$ as a $\mathbb{Q}$-linear combination of the basis but I don't know how to proceed.
$2$) Since $K$ is the minimal subfield in which $O$ can be embedded (by definition of field of fractions), then $O$ is not contained in $K_0$ and thus there exists $\alpha \in O-K_0$.
Any suggestions?
You know $\mathbb{Q}O_L=\bigcup_n \frac1nO_L=L$ for any number field $L$ (Recall the usual proof: for $\alpha\in L$ look at the minimal polynomial $f(X)=a_dX^d+a_{d-1}X^{d-1}+\dots+a_0\in\mathbb{Z}[X]$ of $\alpha$. Then $a_d\alpha\in O_L$ since it satisfies $(a_d\alpha)^d+a_{d-1}a_d(a_d\alpha)^{d-1}+\dots+a_0a_d^d=0$.). With the same notation we have $\alpha^{-1}\in a_0^{-1} O\subseteq\mathbb{Q}O$ (for $0\neq\alpha\in O$) since $a_0\alpha^{-1}=-a_1-a_2\alpha-\dots-a_d\alpha^{d-1}\in\mathbb{Z}[\alpha]\subseteq O$. Hence $O\subseteq K_0$ gives $K=\mathbb{Q}O\subseteq K_0$, which is only possible if $K_0=K$.