# 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 : , , , , , ,
... , , .

### DETAILS

Reference
 I. Vidav, Algebra, Ljubljana: Mladinska knjiga, 1972 pp. 328–330.

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