On 5/8/2012 8:23 AM, Invisible wrote:
>> I should clarify, I took the fact that the extension wouldn't be a field
>> to be implicit in Le_Forgeron's point, since the Hairy ball theorem's
>> use doesn't depend on the numbers forming a field. Not saying your
>> reading was technically incorrect.
>
> By "extend the complex numbers", I implicitly meant to extend it to make
> a larger /field/. Apparently that is impossible. There are of course
> several ways you can extend them if you don't mind the result failing to
> be a field...

This is certainly true, but your options are more limited in 3d than in 
4d.  I was just assuming that Le_Forgeron was already well aware that 
quaternions aren't a field.  Operating under this assumption I took his 
point to be that your options for extending the complex numbers to 3d 
are more limited then when extending them to 4d, even though obviously 
neither extension will result in a field.

Your interpretation of Le_Forgeron is of course also consistent with 
what he posted.  However I'm confused how you reconcile this "any 
extension must be a field" view with your original post.  In the initial 
post you say:

"Question: Why can't you extend the complex numbers to 3D space?

Answer: Because the hairy ball theorem says so."

But that seems a pretty weak explanation of why such an extension isn't 
possible if you're insisting that the extension be a field.  After all 
such an extension isn't possible to 4d, even though the hairy ball 
theorem *doesn't* apply in that case.  If you're going to take this view 
why bother with the hairy ball theorem at all?  You can prove more by 
other means.

If, on the other hand, you allow the extension to not form a field, then 
it makes more sense why you'd use the hairy ball theorem, as it helps to 
explain why the quaternions are possible in 4d but not 3d.  Not saying 
your view is wrong in a technical sense... it just seems a little 
hodge-podge to me.  Perhaps I am missing something?

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