# Ring of Algebraic Integers

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## Theorem

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

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

Let $\mathbb A$ 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 $\mathbb A$ 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 Integral Closure is 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 $K$.

$\blacksquare$