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We all know what the complex numbers are : basically $$R^2$$ with a specified product formula, which is $$(a,b).(c,d)=(ac-bd,ad+bc).$$Hamilton defined the quaternions, which are a wonderful set of numbers, defined to be four-dimensional numbers, which too have a nice formula for products. One can prove the four square theorem using the Hurwitz integers and these numbers have a whole lot of marvellous properties. Can we define three dimensional numbers, who too have nice multiplication and interesting properties?
Also, when I think about algebraic number theory, I imagine rings and fields that are subsets of the complex numbers, whereas here we see how the quaternions have applications in number theory, and they are an extension of the complex numbers. What can we do when we don't restrict ourselves strictly to the complex numbers?
Note: one of the reasons the quaternions are a bit uncomfortable is that they don't commute with each other. However, we still have nice factorization properties when examining the Hurwitz integers. How do they arise? Can we define them as something like the ring of integers of a field?
EDIT:here are some links that give related information: Is there an algebraic closure for the quaternions? and Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work).