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# Gaussians as Euclidean Domain

Follow up to a comment by @nbruin in a previous question:

One reason that Euclidean division isn't available by default on ZI is because as far as sage is concerned, it's a quadratic ring, and those generally are not euclidean rings. There are some quadratic rings that are, but most of them are only euclidean with rather obscure euclidean norms. The fact that Z[i] is euclidean with the "standard" norm is really quite anomalous among quadratic rings.

If you're interested in studying euclidean rings you probably should write some utility functions yourself to help you with it (or search if such utilities are already available). You could even consider writing a new ring subclass for Euclidean rings. To illustrate that sage doesn't know that ZI is a euclidean domain:

sage: ZI in EuclideanDomains()
False


context: ZI = QuadraticField(-1, 'I').ring_of_integers()

My question is: Is there a built in way in SageMath to work with Gaussian integers as an Euclidean domain?

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This is an ongoing process, see trac ticket 23971.

If you are used with Sage development, you can try to apply the changes on your version of Sage and recompile. Otherwise, you can define your own class that inherits from Sage's one and overwrite the _floordiv_ method.

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Asked: 2018-11-13 10:48:43 -0600

Seen: 119 times

Last updated: Nov 13 '18