$\mathcal{O}_K$ analogous to $\mathbb{Z}$? The definition of $\mathcal{O}_K$ isn't very well explained or motivated in my textbook.

Let $K$ be a field a field with $\mathbb{Q} \subset K \subset \mathbb{C}$. $\mathcal{O}_K$ consists of all elements of $K$ which are roots of a normed polynomial over $\mathbb{Z}$, that means
  $$x^n + s_{n-1} x^{n-1} + \cdots + s_1 x + s_0 = 0 \quad \text{ for } n \in \mathbb{N}, s_i \in \mathbb{Z} \, .$$

Now, later this theorem is proven:

For any element $x \in K$ there is an $n\in\mathbb{N}$ with $n x$ in $\mathcal{O}_K$.

From which a corollary "$K$ is the field of fractions of $\mathcal{O}_K$" follows.
But is $\mathcal{O}_K$ the smallest subring of $K$ which has this property? IMHO only then $\mathcal{O}_K$ can really be called analogous to $\mathbb{Z}$ in $\mathbb{Q}$, but this wasn't proven.
 A: What you are referring to is the so called ring of integers in algebraic number theory. The set up is as follows: 
You have $\mathbb{Q}\subset K$ as a finite field extension. Inside of $K$, let $\mathcal{O}_{K}$ be the integral closure of $\mathbb{Z}$. Then you have:
$$
\mathbb{Z}\subset\mathcal{O}_{K}\subset K
$$
as well as $K$ is the fractional field of $\mathcal{O}_{K}$. $\mathcal{O}_{K}$ is the smallest ring inside of $K$ with this property(subring of $K$ and fractional field is $K$), because the rank $\mathcal{O}_{K}$ as a $\mathbb{Z}$ module is the same as $K:\mathbb{Q}$, otherwise we will be missing elements in $K$. So we at least need a subring of $K$ with the same rank as $\mathcal{O}_{K}$ over $\mathbb{Z}$. But then we would not have a good criterion to compare $\mathcal{O}_{K}$ with a different subring of $K$ unless one is inside of the other. However we know $\mathcal{O}_{K}\cong \mathbb{Z}^{n}$ additively, so its only subring (not ideal) of rank $n$ must be itself. There may be other ones inside of $K$ overlap but not inside of $\mathcal{O}_{K}$ as Qiaochu pointed out. 
By the way, it make no sense to say $\mathbb{Z}$ is the "smallest ring with this property". There is no strict subring of $\mathbb{Z}$. 
