Ring of integers is not a field

Question

Let $K$ be a finite extension of $\mathbb{Q}$ (i.e. $K$ is a number field). Let $\mathbb{B}$ be the set of all algebraic integers. Inside $K$, we have the so-called the ring of integers $\mathcal{O}_{K}=K\cap\mathbb{B}$.

It can be checked that $\mathcal{O}_{K}$ is a ring. I strongly suspect that $\mathcal{O}_{K}$ is not a field. Is there any easy way to see this?

Answer 1

$2 \in \mathcal{O}_K$. $\frac{1}{2}$ is not an algebraic integer, and therefore not in $\mathcal{O}_K$.

Answer 2

Hint: $\mathcal{O}_K$ is an integral extension of $\mathbb{Z}$. If $R\subseteq S$ is an integral extension, then $R$ is a field iff $S$ is a field.