Ring of Algebraic Integers

Theorem
Let $K / \Q$ be a number field.

Let $\Z \sqbrk x$ denote the polynomial ring in one variable over $\Z$.

Let $\OO_K$ denote the set of all elements of $K / \Q$ which are a root of some monic polynomial $P \in \Z \sqbrk x$.

That is, let $\OO_K$ denote the algebraic integers over $K$.

Then $\OO_K$ is a ring, called the Ring of Algebraic Integers.

Proof
This is a special case of Integral Closure is Subring.

We have an extension of commutative rings with unity, $\Z \subseteq K$, and $\OO_K$ is the integral closure of $\Z$ in $K$.

The theorem says that $\OO_K$ is a subring of $K$.