Field of Fractions of Number Ring is Corresponding Number Field

In the case of monogenic number fields we have simply that $K(\mathbb{Z}[\alpha ])=\mathbb{Q}(\alpha )$.

Proof

Let $\mathbb{L}=K({\mathcal{O}}_{\mathbb{K}})$. Then, since $\mathbb{L}$ is the smallest field containing ${\mathcal{O}}_{\mathbb{K}}$, and $\mathbb{K}$ is a field containing ${\mathcal{O}}_{\mathbb{K}}$, we have that

Now let $\alpha \in \mathbb{K}$ be arbitrary. Since $\mathbb{K}$ is a number field, there exists an integer $k$ such that $k\alpha \in {\mathcal{O}}_{\mathbb{K}}$ by this result.

So by the above inclusion, we have that $k\alpha \in \mathbb{L}$, and since $k\in \mathbb{Z}\subseteq {\mathcal{O}}_{\mathbb{K}}\subseteq \mathbb{L}$, $k^{-1}\in \mathbb{L}$ from the properties of a field. Hence $k^{-1}k\alpha =\alpha \in \mathbb{L}$. Since we started by assuming $\alpha \in \mathbb{K}$ arbitrarily, we have that $\mathbb{K}\subseteq \mathbb{L}$ and therefore