Euclidean Algorithm in the Ring of Algebraic Integers Generated by the Square Root of Five

This Demonstration shows examples of arithmetic operations in the field , the field of numbers , where and are rational numbers. But instead of using the numbers 1 and , here we use 1 and the golden ratio .

First define the conjugate of to be . Also define the norm of to be .

An algebraic integer in the field is of the form , where . If a number is an algebraic integer, its norm is an ordinary integer.

Write as , with , , and integers.

So .

Suppose that and are algebraic integers in . In the field , the quotient can be written as , where and are rational. Let be such that and . Then . If then , where .

In the case of division in algebraic integers, we show the pair . This Demonstration shows the Euclidean algorithm in the ring of algebraic integers of :