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

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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 :

, ,

, ,

, ,


, ,



Contributed by: Izidor Hafner (February 2020)
Open content licensed under CC BY-NC-SA




[1] I. Vidav, Algebra, Ljubljana: Mladinska knjiga, 1972 pp. 328–330.

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