Integral Multiple of an Algebraic Number

Theorem
Let $K$ be a number field and $\alpha \in K$.

Then there exists a positive $n \in \Z$ such that $n \alpha \in \mathcal O_K$.

In this context, $ \mathcal O_K$ denotes the algebraic integers in $K$.

Proof
If $\alpha = 0$ then any integer works and we are done.

Let $\alpha \ne 0$.

Let $f \left({x}\right) = x^d + a_{d-1}x^{d-1} + \ldots + a_0$ be the minimal polynomial of $\alpha$ over $\Q$.

Suppose that $a_i = \dfrac{b_i}{c_i}$ is a reduced fraction for each $i$ such that $a_i\neq 0$.

Let $n$ be the least common multiple of the $c_i$, of which there must be at least one by our assumptions.

Consider the polynomial:

Note that $g$ is a monic polynomial with coefficients in $\Z$ by our choice of $n$.

Furthermore, by construction, we see that $n\alpha$ is a root of $g$ and is therefore an algebraic integer.