>> Aren't quaternions non-associative?
> 
> Non-commutative you mean, but yes.

Oh. Maybe I'm thinking of hypercomplex numbers then? I recall that it's 
not possible to generalise the complex field to 3 or 4 dimensions 
(although I have no idea why), so all the generalisations people have 
come up with fail to be fields.

By complete coincidence, I spent my afternoon today reading about magmas 
and groups and rings and quasigroups and semigroups and groupoids and 
rings and skew rings and splitting fields and fields of factors and 
polynomial rings and that whole zoo of other things... Apparently 
"vector field" doesn't mean what you'd expect at all! :-D

I've spent a while looking at group theory. It's not "hard", it's just 
that there are lots and lots of terms to learn, mostly with unintuitive 
names, and the whole system of groups constructed from groups 
constructed from groups gets confusing quite fast. Similarly, the stuff 
with rings and fields and so forth isn't "hard", there's just lots of 
stuff to remember...

> Also forgot to mention that (if I recall correctly) ideas related to 
> this play an important role in some of the operations in computer 
> algebra systems.

Mathematica directly uses the commutivity and associativity of various 
operators, not to mention distributivity. Indeed there are built-in 
methods for telling Mathematica that your new function is associative 
and/or commutative. (Distributive requires explicit transformation rules...)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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