The field of fractions of an order is $\mathbb{Q}$ Let $K$ be an algebraic number field and $L$ the algebraic integers in $K$. Assuming an order $O$ is defined as a subring of $L$ that is of finite index as a subgroup of the additive group $L$, I have to prove that the field of fractions of $O$ is $K$.
It is not hard to prove that $O$ is a $\mathbb{Z}$-module of rank $n = [K : \mathbb{Q}]$. Moreover, we know that the field of fractions of $L$ is $K$, which implies that the field of fractions of $O$ is a subfield of $K$. How can we prove that the field of fractions of $O$ is also $K$?

For any element $k = \frac{x}{y}\in K$ where $x, y \in L$, we have $k = \frac{dx}{dy}$ where $d = [L:O]$ is the index. And $dx$ and $dy$ are elements of $O$.
