Let $K$ be a number field. I want to show that $\mathbb{Q}O\_K=K$.

$\mathbb{Q}O\_K \subseteq K$ is clear. Now let $\alpha \in K$, then there exists the minimal polynomial $f(X)=X^n +\cfrac{a\_{n-1}}{b\_{n-1}}X^{n-1}+\dots+\cfrac{a\_1}{b\_1}X + \cfrac{a\_0}{b\_0} \in \mathbb{Q}[X]$. We need to find some $g(X)\in \mathbb{Z}[X]$ such that $g(\alpha)=0$ so that $\alpha \in O\_K$. 

(I do not know what $\mathbb{Q}O\_K $ is actually, I am guessing that it consists of elements $q\beta$, i.e., $q\in\mathbb{Q},\beta \in O\_K$. )
So, let $D=lcm(b\_0,b\_1,\dots,b\_{n-1})$, then $Df(X)= f(X)=DX^n +D\cfrac{a\_{n-1}}{b\_{n-1}}X^{n-1}+\dots+D\cfrac{a\_1}{b\_1}X + D\cfrac{a\_0}{b\_0} \in \mathbb{Z}[X]$ and $Df(\alpha)=0$ however it is not monic so I can not conclude that $\alpha \in O\_K$. How can I solve this problematic part and what is $\mathbb{Q}O\_K$ exactly?