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Division in the Ring of Algebraic Integers Generated by the Square Root of Five

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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 .

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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 .

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

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Reference

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

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