Extended GCD of Quadratic Integers

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Consider the quadratic field and the associated ring of integers , where if and if . We assume is principal but not necessarily Euclidean. We compute the GCD of two elements , of modulo a unit of . The computation also gives explicit coefficients , for the Bézout identity . This is done by reducing binary quadratic forms and considering the sum of ideals as the ideal , with .

Contributed by: Abdelwaheb Miled and Ahmed Ouertani (March 2011)
Open content licensed under CC BY-NC-SA



This algorithm computes the GCD of two quadratic integers; it does this by combining the following operations:

1) compute the sum of two ideals of and put the ideal result into canonical form: , , so that the norm of this ideal is then .

2) find a generator of an ideal (an element such that is equal to the norm of the ideal, that is, ). Recall that the norm of an element is , where if and if .

3) represent the integer 1 by a binary quadratic form.

This method is quite efficient for small quadratic integers, but could be lengthy for large numbers.


[1] D. A. Buell, Binary Quadratic Forms, New York: Springer–Verlag, 1989.

[2] A. Miled and A. Ouertani, "Extended GCD of Quadratic Integers," arXiv, 2010.

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