Let K be a number field of extension degree d over Q. Then an order O of K is a subring of the ring of integers of K with d generators over Z, including 1. The ring of integers of every number field K is an order, known as the maximal order, of K. Every order of K is contained in the maximal order. If α is an algebraic integer in K, then Z[α] is an order of K, though it may not be maximal if d is greater than 2.

algebraic integer | extension field degree | field order | number field | order | ring | ring of integers