Ring of integers of a number field

The ring of integers of a number field $K$ is

\mathcal O_K=\{\alpha\in K:\alpha\text{ is an algebraic integer}\}.

Equivalently, it is the integral closure of $\mathbb Z$ in $K$ . Addition and multiplication preserve integrality, so this subset is a subring with identity.

Size and coordinates

As an additive group it is free of rank $[K:\mathbb Q]$ . A choice of its integer coordinates is an integral basis. This does not assert that it has a basis of powers of one element.

Examples

One has $\mathcal O_{\mathbb Q}=\mathbb Z$ and $\mathcal O_{\mathbb Q(i)}=\mathbb Z[i]$ . For $\mathbb Q(\sqrt{-3})$ , the element $(1+\sqrt{-3})/2$ is also integral, so merely adjoining the square root misses part of the ring.

References

J. S. Milne, Algebraic Number Theory, v3.08. Author’s text, §2, Proposition 2.29 and Corollary 2.30.