Order (reviewed)

An order in a number field $K$ is a subring of $K$ which is also a lattice in $K$. Every order in $K$ is contained in the ring of integers of $K$, which is itself an order in $K$; for this reason, the ring of integers is sometimes called the maximal order.

Example: $\Z[\sqrt{5}]$ is an order in $K=\Q(\sqrt{5})$. However, it is not maximal, since the maximal order (i.e. ring of integers) of $K$ is $\Z\left[\frac{1+\sqrt{5}}2\right]$.