Division in the Ring of Algebraic Integers Generated by the Square Root of Five
This Demonstration illustrates division in the ring of algebraic integers in the field , that is, the field of numbers , where and are rational. But instead of using the numbers and , here we use and the golden ratio .[more]
An algebraic integer in the field is of the form , where . Write as , with , and integers.
The conjugate of a number is . The norm is defined by . So . If a number is an algebraic integer, its norm is an ordinary integer.
Suppose that and are algebraic integers in . The quotient can be written as , where and are rational. Let be such that and . Then . If then , where .[less]
 I. Vidav, Algebra, Ljubljana: Mladinska knjiga, 1972 pp. 328–330.