Ring of Algebraic Integers

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

Let $\mathbb A$ denote the set of all elements of $K / \Q$ which are the root of some monic polynomial in $\Z[x]$.

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

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

Proof
This is a special case of the integral closure being a subring.

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

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