2. Extending the Rationals with the Square Root of Five

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This Demonstration shows examples of arithmetic operations in the extended field , that is, in the field of numbers , where and are rational numbers. But instead of using the numbers and , here we use 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. Now we want to get the quotient in the ring of algebraic integers. The procedure is this: Let , where and are ordinary integers such that and . Then . The reminder is . So we write , where . The button "division in integers" shows the pair , the quotient and reminder.


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


Algebraic integers are roots of polynomial equations , where coefficients are integers. Algebraic integers in of the form satisfy the equation .


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


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