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
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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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