GCD over Lipschitz quaternions is discussed in detail in http://arxiv.org/abs/1202.3198 Left and right GCD are distinct functions, defined up to a (right and left resp) unit factor, in {1, -1, i, -i, j, -j, k, -k} . WFL On 11/28/13, Henry Baker <hbaker1@pipeline.com> wrote:
I don't have my Knuth in front of me, but I think Knuth already talked about the non-commutative case, including matrices of various sorts.
Also, doesn't Hurwitz talk about gcd's over his quaternions?
At 12:05 PM 11/28/2013, Dan Asimov wrote:
Anyway, I think GCD makes the most sense only in principal ideal domains.
Hmm, I think we've been tacitly assuming the ring is commutative. What if it isn't?
Take for example the ring Li of Lipschitz quaternions := Z + Zi + Zj + Zk.
Or the ring Hu of Hurwitz quaternions := Li[(1+i+j+k)/2].
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