Quasifields and division rings:

(Possibly more interesting than the title would suggest. Possibly not)

I'm in the middle of proving that you can have something that is a quasifield but not a division ring. This may sound almost like a semantic game, but it's actually kind of cool. See, a quasifield is a set under two operations which satisifes five axioms.

• Q forms a commutative group under addition.
  
• If we have any two nonzero of x,y,z, the equation xy=z has a unique nonzero solution.
  
• basically that we have identities
  
• (a+b)m=am+bm
  
• r,s,t \in Q and r!=s. Then xr=xs+t has a unique solution.

A division ring is a ring with no zero divisors, and it's useful because only in a division ring can we think of, well, dividing. If you have zero divisors, it quickly goes to hell.

A quasifield is actually also useful. It turns out that every plane that does what we (and Euclid) want a plane to do can actually be formed from a quasifield in a not very difficult procedure (basically, you define lines to be of the form x=c and y=mx+b for c,m,b in our quasifield). The xy-plane can be formed this way over R. But it's true that any plane can be formed this way, giving us finite planes and other bits of weirdness I'm still wrapping my head around.

It is interesting to me that a quasifield doesn't have to be a division ring, though. Basically, it comes down to saying that right distributivity doesn't have to hold in a quasifield. It seems a bit surprising that there would be something special about left distributivity; it seems that the definition of right or left is completely arbitrary. But it is possible to construct something that basically looks pretty nice in which right distributivity doesn't hold, and what's more, to be useful in geometry, we require left distributivity but not right distributivity.

Isn't that a bit counter-intuitive?

Anyway, the question that I have (and the problem that I'm having recently with math), is why do I care? I mean, I think affine planes (which are planes that behave how we intuitively expect planes to behave) are kind of cool, and I see why studying quasifields is important in looking at affine planes. But what, really, do these strange planes matter? It's not like I will ever see a finite plane?

I can't believe that I am no longer championing knowledge for its own sake. I mean, I still sort of am, but I can no longer practice what I preach. I'm glad that there are people who learn stuff for the sake of learning it, but I need more concrete reasons. I want to learn something because it's useful rather than because it's pretty. And I cannot believe that I'm writing that, because it's quite a turn-around for me. But there you have it. It's too late for them to kick me out of the U of C, right?