9873

Extended GCD of Quadratic Integers

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 .

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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.
References:
[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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